LIBRARY OF CONGRESS. 



Shelf-- 

UNITED STATES OF AMERICA. 



UKV 



IOC 



TEIGONOMETRY: 



ANALYTICAL, PLANE AND SPHERICAL. 



WITH 



LOGARITHMIC TABLES. 



BT 



DE YOLSON WOOD, 

PROFESSOR OF MATHEMATICS AND MECHANICS IN STEVENS' 
INSTITUTE OF TECHNOLOGY. 






MAY 7 188- 



o tT jo$SH 



NEW YORK: 

JOHN WILEY & SONS, 

15 Astor Place. 

1885. 



»1 



Copyright, 1885, 
By DE VOLSON WOOD. 



Press of J. J. Little & Co., 
Nos. 10 to 20 Astor Place, New York. 



PREFACE. 



An author of a new trigonometry, at the present time, 
owes an explanation if not apology, both to teachers 
and students, for adding another to the already too 
numerous works upon this subject. The author was 
frequently applied to for opinions upon works extant 
and those proposed for publication, but found none 
that quite suited him ; so the present work was under- 
taken to provide for his own classes and avoid further 
annoyance in being called upon to criticise others, even 
though it furnishes one more book for other critics to 
pass upon. The author found, however, when he had 
reduced his ideas to a form ready for print, that he did 
not differ from certain other writers as much as he, at 
first, expected to ; still the many distinctive character- 
istics which remain will, we trust, commend themselves 
to others. 

Attention is called to the fact that the trigonometri- 
cal functions are first defined and treated as ratios, but 
that afterwards they are represented by lines, which 
lines are so defined as to represent the functions. The 
latter may be made more prominent by any instructor 



iv PREFACE. 

who desires so to do, although the chief object in the 
plan of the work was to furnish an aid to the memory 
for those who can remember geometrical representations 
more easily than abstract ratios. 

We trust that the new notation in article 32 will 
meet with favor, as its adoption will remove an am- 
biguity which has thus far existed in this science. The 
functions as defined in article 16 are not restricted to 
quadrants, and their treatment afterwards in articles 
23 to 30 are perfectly general, and hence it seems un- 
necessary after deducing equation (30) to make other 
and special demonstrations for particular limitations of 
the angles x and y. Similar remarks apply to equa- 
tions following equation (30). Imaginary, or impossi- 
ble functions have been introduced for the purpose of 
affording a greater variety in the exercises. 

AUTHOR. 
Hoboken, 1885. 



CONTENTS 



CHAPTER I. 

GENERAL PRINCIPLES. 

ARTICLE PAGE 

1. Definitions 1 

2. Angle 1 

3. Constant ratios 3 

4. Notation 5 

6. A function 7 

7. Exercises , p 8 

8. Analytical trigonometry 10 

9. Signs of arcs and angles 10 

10. Angles unlimited 11 

12. Coordinates 12 

13. The abscissa 12 

14. The ordinate , 13 

16. General definitions of trigonometrical functions 14 

17, Signs of the functions 15 

19. Limiting values of the functions 16 

20. Relative values of the functions 16 

Exercises 19 

22. Functions of negative angles 20 

23. The complement of an angle 21 

24. The supplement of an angle 21 

25. Complementary functions 21 

26. Functions of 90° + y 23 

Exercises 24 

27. Functions of 180° - y 25 

30. Trigonometrical lines 27 

Exercises 30 

32. Circular functions 31 



yi CONTENTS. 

ARTICLE PAGE 

Exercises 33 

34. Multiple angles 34 

35. Impossible values 35 

36. The sinusoid 36 



CHAPTER II. 

OF THE FUNCTIONS OF THE SUM AND DIFFERENCE OF ANGLES. 

37. The sine of the sum of two angles t 37 

38. The cosine of the sum of two angles 38 

Exercises 39 

39. Discussions of equations 41 

40. The tangent of the sum of two angles 43 

42. Functions of double the arc 44 

43. Functions of an angle in terms of half the angle 44 

48. Construction of tables 53 



CHAPTER m. 

SOLUTION OF PLANE TRIANGLES. 

49. Eight angled triangles 54 

51. Given two angles and one side 61 

52. In an oblique triangle given two sides and an angle opposite 

one 62 

53. In an oblique triangle given two sides and the included angle. 65 

54. In an oblique triangle given the three sides 67 

55. Solution of oblique triangles by means of right triangles 70 

57. Area of triangles 74 

Practical examples 78 

61. Spherical geometry 82 

CHAPTER IV. 

OF SIGHT ANGLED SPHERICAL TRIANGLES. 

62. Formulas for right triangles 85 

64. Napier's circular parts 87 

65. Napier's rules 88 



CONTEXTS. vii 

ARTICLE PAGE 

66. Species of the parts 89 

67. Given the hypothenuse and a side £0 

68. Given one angle and side opposite 92 

69. Given the hypothenuse and an angle 93 

70. Given an angle and a side adjacent 93 

71. Given the two sides 94 

72. Given the two oblique angles 94 

73. Quadrantal triangle 94 



CHAPTEE V. 

OBLIQUE TRIANGLES SOLVED BY EIGHT TRIANGLES. 

75. Sines of the angles proportional to the sides opposite 96 

77. Given two sides and included angle 98 

78. Given two angles and the included side 98 

79. Given two sides and angle opposite 99 

80. Given two angles and a side opposite 99 

81. Given the three sides. ... 100 

82. Given the three angles 100 



CHAPTER VI. 

GENERAL FORMULA. 

83. Cosine of a side of a spherical triangle 102 

84. Spherical triangle and its polar 103 

85. Value of sin a cos B, etc 104 

87. Formulas adapted to logarithmic computation 105 

90. Napier's analogies 108 

91. Given two sides and the included angle 110 

92. Given two angles and included side Ill 

93. Given two sides and an angle opposite 112 

94. Given two angles and side opposite 113 

95. Given the three sides 113 

96. Given the three angles 113 

97. Area of a spherical triangle 113 

98. Definitions of astronomical terms 114 



v iii CONTENTS. 

ARTICLE PAGE 

99. Shortest distance between two places on the earth's surface. . 116 

100. Latitude of a place 117 

101. To find the time of sunrise 117 

102. To find the time of day 119 

103. To find the azimuth of a star 120 

104. To find the common azimuth of two stars 120 



Explanation of the tables.. 129 

Logarithmic tables of numbers and of trigonometrical func- 
tions 139 

Tables of natural trigonometrical functions 201 



TRIGONOMETRY. 



CHAPTEE I. 

FUNDAMENTAL PRINCIPLES. 

1. A geometrical quantity is any form, figure, or mag- 
nitude conceived of in space. 

A geometrical point is a figure from which magnitude 
is abstracted. It has no size and simply marks a place. 

A geometrical line is that property of a figure which 
has length only ; or from which breadth and thick- 
ness are abstracted. 

A line may be generated by the movement of a point. 
If the generating point preserves a constant direction, 
the line generated is a right line, but if it changes its 
direction at every point, it generates a curved line. A 
right line is always understood unless otherwise stated. 
Aright line is said to be given when its length and 
direction are known in reference to some other line 
assumed to be fixed. 

2. Angles. An angle is the difference in direction of 
two lines. It is measured by means of some angle 
arbitrarily assumed for the unit. In reference to a 
figure representing an angle, it may be considered as 
the opening between two lines; the opening being 
greater as the inclination between the lines is greater. 

1 



TRIGONOMETR T. 



Thus the direction or inclination of the line OF 
compared with that of OE is 
the angle between them. An 
angle is read by means of letters 
as EOF, the letter at the 
vertex being the middle letter; 
but sometimes it is referred to 

by simply naming the letter at the vertex as 0. 

An angle may be denoted by a single letter, as x, y, 

etc., or by a Greek letter as a, fi, y . . . 6 9 cp, etc., 

in which case we have 




x = EOF, 



or <p = EOF, 



etc. 



An angle may be generated by revolving a line about 
a point. Thus, the angle EOF may be conceived to be 
generated by revolving a line from OE into the final 
position of OF. The line OE from which OF starts in 
the rotation, is called the initial Urn, and OF in its 
final position, the terminal line. The moving line, OF, 

is called the generatrix. The 
point is the origin, vertex, or 
pole. An angle in this case 
measures the amount of rota- 
tion of the generating line. 

If two lines do not intersect, 
they are conceived to make the 
same angle as two other lines 
passing through a point and parallel respectively to 
the given lines. Thus, the angle between the lines 
AB and" CD, whether in the same plane or not, is the 
same as between the lines OE and OF, drawn through 
the point and parallel respectively to CD and AB. 




FUNDAMENTAL PRINCIPLES. 



3. Measurement of Angles. If two lines, AB and 
CD, so intersect as to make the four angles at equal, 
the lines will be mutually perpen- 
dicular, and eacft%f the angles is 
called a right angle. A right angle 
is sometimes taken as a unit angle. 
An acute angle is less than a right 
angle, and an obtuse angle greater. 
An angle may be measured by 
the arc subtending it, the centre 
of the arc being at the vertex of 
the angle. For, the radius being constant, the angle and 
arc will be generated by the same amount of rotary 

motion of the generating line 
OB, and of the point B. In 
order to compare angles by 
means of arcs, the radius must 
be constant, and, unless other- 
wise stated, it will be considered 
as unity. Then will n be the 
length of a semi-circumference ; 
and \tt of one-fourth of a cir- 
cumference and hence will subtend a right angle, n 
will subtend two right angles, \tt 
three right angles and, generally, 
\n7t^ n right angles, n being any 
integer. We say, for brevity, \it 
is a right angle, \n is one-half of 
a right angle, etc. This method of 
measuring an angle is called cir- 
cular measure. 

If 360 lines radiate from a point, 
making equal angles between the consecutive lines, each 





4 TB1G0JX0METBY. 

angle is, by common consent, called a degree. This is 
the unit-angle commonly employed in practice. 

Hence, we say, that the complete revolution of a line 
about a point generates 360", and that in one right angle 
there are 90 degrees. Also an arc \n subtends an 
angle of 90 degrees, \n of 45 degrees, and so on. And 

for brevity we sometimes 
say, \n equals 90 degrees, 
\ti equals 45 degrees, etc. 

An angle may also be 
measured by means of cer- 
tain other lines bounding 
it. Thus, if from any points B\ B, B ', etc., in the line 
OB, perpendiculars, B'A r , BA, B"A\ etc., be let fall 
upon OA we have from similar triangles 




A'B ' _AB_ A'B" 
OA ~ OA~ OA' 



= a constant 



for any fixed angle ; hence, if this ratio be determined 
in any manner, it may be used as a measure of that 
angle. Thus, this ratio for an angle A OB — 45 degrees, 
will be unity ; hence, conversely, when this ratio is 1 we 
know that the angle is 45 degrees. The ratio for 
other angles may be found and the results tabulated. 
We also have 



OA 
OB 



OA 
OB 



a constant 



for a given angle. Similarly, ratios may be found 
between other lines about the angle. This mode of 
measuring angles forms the basis of the science which 
we are to consider. 



FUNDAMENTAL PRINCIPLES 7 

(sin A) n , or sin n A ; the latter being the more com- 
mon notation ; and similarly for other functions. The 
sine of the nth. power of the angle is written sin A n ; 
and similarly for the other functions. The sine of 
n times an angle is written sin (nA) , or simply 
sin nA ; and similarly for the other functions. The 
reciprocal of the sine of A is written (sin^) -1 or 

- 7, but not sin~ 1 A , the latter notation being 

sin A 

offcen employed for another purpose. The functions 
may be multiplied and divided in the usual manner ; 
thus sin A cos A implies that cos A is multiplied by 
sin A, and similarly for the others. 

6. One quantity is said to be a function of another 
when it is so related to the other that a change in the 
value of the latter causes a change in the value of the 
former, thus in the expressions 

y = 3x? — ax, y — log x, y = sin x, y — tan x, 

y is said to be a function of x; and the first is called an 
algebraic function, the second is logarithmic, and the 
others trigonometrical. All functions not algebraic are 
called transcendental When the form of a function is 
unknown, or when brevity alone is desired, it may be 
written 

y = F (x), or y =/ (x), or y = <p (x) ; 

all of which are read "y is a function of a?." In these 
expressions F, f, (p, are not quantities but symbols 
implying some form of an expression. 

7. Trigonometry is that science which treats of the 



8 



TRIGONOMETRY. 



properties and relations of trigonometrical functions, 
and of their use in the solution of triangles. It is con- 
sidered in three parts, viz. : 

Analytical Trigonometry treats of the abstract prop- 
erties of trigonometrical functions. 

Plane Trigonometry treats of the solution of plane 
triangles by means of trigonometrical functions. 

Spherical Trigonometry treats of the solution of 
spherical triangles by means of trigonometrical func- 
tions. 




EXEKCISES. 

1. In the right triangle BAG, if BA = 4, AC = 3, what will be the 

sine of the acute angles ? 

[We have BC = V4FT~& = 5 ; 
hence, according to the defini- 
tion, sin B — J and sin C = £] 
2. Find the value of the other seven 
trigonometrical functions of B 
in the preceding exercise. 

3. Find the trigonometrical functions of 45°. 

In this case BA - AC, and BC = BA V2. It will be found that 
sin 45° = $ V% — cos 45° ; tan 45° = 1 = cot 45° ; sec 
45° = 4/2 = esc 45°. 

4. Find the trigonometrical functions of 30°. 

Let CBD be an equilateral triangle, then will each 
of the angles be 60°. From B drop the perpen- 
dicular BA, then will BAC = 90°, CBA = 30% 
AC=AB = \CD = \BC\ 




AC 



BA 



tan 30 c 



sin 30° = 


BC 


BC 


= i ; 


cos 30° 


~ BC 


VBC*- 
BC 

— yr. 

— r 8 1 


AC* V4.AC 2 - 
2 AC 
cot 30° = YW. 


AC _ 


:*V3"; 



FUNDAMENTAL PRINCIPLES 9 

5. Find the trigonometrical functions of 60°. 

UB= 30° in the right triangle BAG, then will C = 60°. 

A T> 

sin 60° = ==5 = i VS = cos 30° ; cos 60° = k = sin 30° ; 
tan 60° = VS = cot 30° ; cot 60° = J VW= tan 30°. 

6. Show that sin 2 A + cos 2 A = l. 

[We have from group (A) 

sin* A + cos 2 ^ =■£. + p =^p^ = p = 1-] 

7. Show that sec 2 A = 1 + tan 2 A 

8. Show that 



tan J. 



sin .4 _ sin ^ _ 1 

cos J. — Vl — sin 2 J. ~~ Vcsc 2 A — 1 * 



Prove the following equations : 
9. tan A cot A = 1. 

10. cos J. = sin ^4 cot A. 

11. (sin J. + cos Af 4- (sin J. — cos Af = 2. 

12. sin 2 J. - cos 2 B = sin 2 5 - cos 2 A. 

13. tan J. + cot A = sec A esc J.. 

14. (sin 2 6 4- cos 2 0) B = 1. 

15. (sin 2 <p — cos 2 cpf — 1 — 4 cos 2 <p -f 4 cos 4 9). (Observe that from 

Example 6 we have sin 2 q> = 1 — cos 2 q>.) 

_,„ tan <p 4- tan , , c 

16. , = tan <z?.tan 0. 
cot <p 4- cot 

17. 3 sin 60° - 4 sin 3 60° = 4 cos 3 30° - 3 cos 30°. 

18. sin 2 (0 4- q>) + cos 2 (0 + ?>) = 1. 

19. sin 45° + cos 30° = i (V2~4- V3). 

20. tan 30° . cos 45° = J. 

21. . ^ — — = (sec <z> — tan q>f. 

1 + sm <p v ^ ™ 

22. sin 60° . cot 60° = h 

23. (tan A 4- cot A) sin A . cos J. = 1. 

24. tan 30° . tan 60° . cos 45° . sec 45 c = 1 

25. cos x . tan x — sin x. 

26. esc x . tan x — sec x. 



10 



TRWONOMETR Y. 



Analytical Trigonometby. 

8. We now consider the principles according to 
which the functions may be so extended as to be more 
general. The algebraic signs + and — applied to a 
line indicate opposite directions. Thus, if a line gener- 
ated by a point moving in one direction be positive, the 
opposite direction will be negative ; the former of which 

a -n will be indicated by + and the 

^ latter by — . If AB be positive 

BA and AC will be negative. 

9. Angles and arcs may also be affected by the signs 
+ and — . Thus, if the angle XOP, generated by a 








n- 



+?k 



left-handed rotation, be considered positive, the angle 
X0P u generated by a right-handed rotation, will be 
negative. The signs + and — indi- 
cate relative direction of motion in 
generating a magnitude. 

The angle POX would also be 
negative, the order of the letters in- 
dicating that the rotation is from 
P to X, or right handed. The posi- 
tive arc is here indicated by -f <p, and the negative arc 
by - <p. 

Since the above relations are arbitrary, either may be 
assumed as positive, but when once chosen it must be 




FUNDAMENTAL PRINCIPLES. 11 

used in that sense through that discussion. According 
to custom, left-handed rotation will be understood un- 
less otherwise stated. 

10. Theoretically \ angles have no limits as to size, for 
the generating line may be revolved about the pole 
any number of times ; still all possible directions in a 
plane will have been passed over by one revolution of 
the radius. 

If cp be an angle less than 360°, the terminal posi- 
tion of the generating line will be the same for the 
angles <7 ! , 2t + cp, 4tt + cp . . . %in + cp, where n 
is zero or an integer either positive or negative. 

11. If OB be the terminal line, 
any point in it, as B, may be de- 
termined by any one of the four 

following ways : By revolving a J [ n/ &\ a iA 

line from OA in a positive direc- 
tion through an angle A OB until 
it passes through the point B, 
when it will be determined by 
the angle A OB and distance OB. Or, by revolving the 
generating line negatively through the angle abc until it 
passes through B. Or, by revolving it positively through 
the angle clef into the position OB', so that if prolonged 
negatively through it will pass through B. Or, by 
revolving it negatively through the angle gh into the 
latter position. 

To represent these algebraically, let p — the distance 
OB, and cp — AOB ; then the point B may be found by 
either one of the four following combinations : — 




f ' -(360 - cp) \ ' + (180 + cp) j ' -(180 - cp) f • 



12 



TRIGONOMETRY. 



12. Coordinates. 


Two] 




Y 




P _______ 


Z> 2 




• \ 
| \ 
\ 




___F? 




^ i 

i 


a, ! \ 




cu !,^iv 






i A 


y 
\ y 

y 
I • 

1/ 
pl_. 


_>3 _^ 

-Y' 


^ 



The negative radius, — p, is unnecessary in this sci- 
ence, and hence will not be further considered, but both 
positive and negative angles will be used. 

Two mutually perpendicular lines 
XX, YY, when used as 
lines of reference, are called 
coordinates, and the lines 
are called coordinate axes. 
The horizontal line XX' is 
called the axis of abscissas, 
and the perpendicular line 
YY' is called the axis of or- 
dinettes. The angle XOY 
is the first quadrant ; YOX, 

the second; XOY, the third; and YOX, the fourth 

quadrant. 

Coordinate axes may be oblique, but as used in this 

work it should be observed that they must be mutually 

perpendicular. 

13. The abscissa of a point is its distance from the 
axis of ordinates YY, measured on a line parallel to 
the axis of abscissas. The abscissas to the right of 
YY will be considered positive, those to the left negative. 
Thus, the abscissa of P 3 will be + b\P\ ; of P 2 , — b%P 2 ; 
of P s , — b 3 P 3 ; and of P 4 , + b±P±. The letters b l9 b. 2 , etc., 
are placed before P l9 P 2 , etc., so that the order of the 
letters will indicate the direction of P l9 P 2 , etc., from the 
axis YY. 

14 The ordinate to any point is its distance from the 
axis of abscissas measured on a line parallel to the axis 
of ordinates. The ordinates above XX will be con- 
sidered positive, those below, negative. Thus, the ordi- 



FUNDAMENTAL PRINCIPLES. 



13 



nate of P x will be + a 1 P i ; of P 2 , + &2P2 \ of P 3 , — a 3 P 3 ; 
of P 4 , — a 4 P 4 . 

The abscissa and ordinate of a point are together 
called its coordinates. 

15. The signs of the coordinates in the four quadrants 
will be 

1st 2d 3d 4th 

Quadrant. Quadrant. Quadrant. Quadrant. 

Abscissa, + — — + 

Ordinate, + + — — 



EXERCISES. 

Show in what quadrant the revolving line will be when it has de- 
scribed the angles, n being an integer : 



1. 

4. 
7. 
0. 


120° 
700° 

f* 

2 n % 






2. 390° 
5. - 120° 
8. 10i* 

11. (2 n + i) it 


3. 
6. 
9. 

12. 


490° 

- 490° 

-8i7T. 

— (4 n — I) it. 


3. 


(6 71 + 


1) 


it 
3 ' 


14. (6 n - 2) | 


15. 


-(8n + 2)|.. 



16. Letting XOP be any angle x, the 
ordinate s, OP = r, the ra- 
dius vector, or simply the 
radius, being the distance 
from the origin to any 
point P in the terminal 
side of the angle x, XP 
= o, the ordinate to the 
point P, TP = a, the ab- 
scissa to the same point, 
where r, o, a, are initial 
letters of the quantities 



origin of co- 




14 



TRIGONOMETRY. 



they respectively represent ; we define the trigonometri- 
cal functions in a more general way, as follows : 



sin x = 



cos x = 



ordinate 
radius 

abscissa 
radius 

ordinate 



ICLJJ. vO 


abscissa 


cot # 


abscissa 
ordinate 


sec x 


radius 
abscissa 


CSC # 


radius 
ordinate 


vsn x 


_ r — abscissa 


radius 


n.xra <v* 


_ r — ordinate 



XP _o 
0P~ r 

YP_a 
0P~ r 

XP_o 
0X~ a 

OX_a 
XP~ o 

OP_r 
'OX~a 

OP _ r 
XP~ o 

r — a 



r — o 



radius 



CB> 



in which o and a may be plus or minus. 

The definitions on page 6, were, strictly speaking, 
for an acute angle, but by giving + and — signs to o and 
a as may be necessary, the definitions in group (B) are 
general, and applicable to any angle from OX to the 
terminal position of the generating line. 

17. Signs of the functions. The signs of the co-ordi- 
nates given in article 15, applied to group (B), gives for 
the signs of the functions for the several quadrants : 



FUNDAMENTAL PRINCIPLES. 



15 



Functions. 


1st 
quad. 


2d 
quad. 


3d 
quad. 


4th 
quad. 


sin and esc 


+ 
+ 
+ 

+ 


+ 
+ 


+ 
+ 




cos and sec 


-f 


tan and cot 




vsn and cvs 


+ 







The sine and cosine have like signs in the 1st and 
3d quadrants, and the tangent and cotangent are posi- 
tive in the same quadrants. 



COS. 



TAN. 




CSC. 



SEC. 



COT. 



18. Limiting values of the ordinate and abscissa. For 
zero degrees, the terminal and initial lines coincide, 
hence the ordinate will be zero, and the abscissa equal r. 
Symbolically, we have for x = 0°, a = r, and o = 0. In 
this way we find from the figure that for 



x= 0°, 


a = 


r, 


o = 


o, 


x= 90°, 


a = 


o, 


o = 


r, 


x = 180°, 


a = 


~ r, 


= 


o, 


x = 270°, 


a = 


0, 


= 


— fj 


x = 360°, 


a = 


r, 


= 


0; 



the last being the same as the first, and after this the 
values repeat themselves at the end of every quadrant. 



16 



TRIGONOMETRY. 



19. Limiting values of the functions. The values 
of a and o given in the preceding article, substituted in 
group (B) give 







Angles. 




Functions. 


















0° 


90° 


180° 


270° 


sin 


TO 


1 


±0 


- 1 


cos 


1 


±0 


- 1 


TO 


tan 


TO 


± oo 


TO 


± oo 


cot 


qp 00 


±0 


4= oo 


±0 


sec 


1 


± oo 


- 1 


T oo 


CSC 


T oo 


1 


± oo 


- 1 


vsn 


+ o 


1 


2 


1 


CVS 


1 





1 


2 



[The order of the signs ± indicates that the function 
has passed from — to + ; that is just before the gener- 
ating line reached the given angle the'sign of the func- 
tion was — , and was + immediately after passing it.] 

20. Relative values of the functions. From group 
(B) we derive the following : 



o 

sin x r o 

= — = — = t an x 

cos x a a 



(i) 



COS X 

sin x 



a 



= cot x ; 



(2) 



that is, The tangent of any angle equals the ratio of the sine 
of that angle to its cosine ; and 



FCXDAHEXTAL PBIXCIPLES. . 17 

The cotangent of an angle equals the ratio of the cosine to 
the sine. 

Also from the same group we find the following com- 
binations of the functions equal to unity : 

cC ~ + ° 2 r * 1 /Q\ 

sin" x -r cos' x — — = — = -= = 1, \6) 

tan x cot a? = 1, (4) 

sin 2 1 esc x = 1, (5) 

cos a; sec a; = 1, (6) 

cos a; + vsn x = 1, (7) 

sin z 4- cvs x = 1. 1^8 ) 

21. Each function may be expressed in terms of any one 
of the other seven. Thus, from equations (3), (5), (8), of 
the preceding article we have directly 

1 



sin x = x 1 — cos' x = = 1 — cvs x, 

CSC X 



which gives the value of sin x in terms of cos, cse, and 
cvs. To find it in terms of the tangent, we have from (1), 

sin x = tan x cos x 
from (3), 



= tan x Vl — sin 2 x ; 
.\ sin 2 x — tan 2 x — tan 2 x sin* x> 

transposing, 

(1 + tan 2 x) sin 2 x = tan 2 x ; 
2 



18 



TRIGONOMETRY. 
tan x 



.*. sin x ■ 



VI + tan 2 x' 



(9) 



which is the required result. 
From (4) we have 

tan x = 



cot #' 
which substituted in the preceding equation gives 

1 



(10) 



sin x : 



VI + cot 2 X ' 



and so on for the other two functions. In this way the 
following table may be formed. 



0Q 

K 2 






Eequired 


Functions. 








sin 


cos 


tan 


cot 


sec 


CSC 




sin 

4/1 - sin 2 
sin 




tan 


1 








4/sec 2 -1 
sec 

1 
sec 


1 


sin 


^1 - cos 2 
cos 




4/1 + cot 2 
cot 






4/1 + tan 2 
1 


CSC 




|/csc 2 - 1 




Yl + tan 2 
tan 

1 

tan 

4/1 + tan 2 


4/I + COL 2 

1 

cot 
cot 


c^c 




4/1 -cos 2 


1 


tan 


4/sec 2 -1 

1 




4/1 - sin 2 ~ 


COS 

cos 


|/csc 2 - 1 




4/1 - sin 2 




cot 


^c*c 2 - 1 




sm 

1 


4/1 - COS 2 

1 

COS 

1 


4/sec 2 - 1 
sec 

sec 




4/1 + cot 2 


CSC 




4/1 - sin 2 

1 
sin 


cot 


Vcsc 2 - 1 




4/1 + tan 2 




CSC 


4/1 + cot 2 


CSC 


yi -cos 2 


tan 


4/sec* - 1 



FUNDAMENTAL PRINCIPLES. 19 

EXERCISES. 

1. In what quadrants may A be if sin A — — 0.567 ? 

2. If sin x = — |, find the other functions when the terminal line is in 

the third quadrant. 

3. Find the functions when tan x = 2, and the terminal line is in the 

third quadrant. 

4. Find x when sin x = cos x. 

[Make cos x — VI — sin 2 x, solve and compare the result with 
exercise 3, page 8.] 

5. Find x when sin x = tan x. Ans. x = 0. 

6. Find the trigonometrical functions corresponding to cot x = 2. 

7. If tan x = 2 cos x, find sin x in terms of cot x. 

["tan x — , and cos x = Vl — sin 2 x ; hence we find 

cot x 



-Vi- 



4 cot 2 x 



] 



8. If cot x = 2 tan x, find sin x in terms of cos x. 

9. Find sin x from the equation 2 sin 2 x — 3 sin x = 1. 

[Here sin x is the unknown quantity, and the equation is to be 
solved as a complete quadratic] 

10. Find tan x from the equation a tan x + b cot x = c. 

[Substitute cot x from equation (4) and solve.] 

11. Find sin x from the equation 3 sin x — 2 cos x = 2 vsn #. 

Ans. |. 

12. Show that sin 2 \x + cos 2 $x = 1. 

13. Show that sin 2 nz = 1 — cos 2 nx. 

14 What does sin 2 (1 — \x) + cos 2 (1 — \x) equal ? 

15. In a right triangle, if the base be 3a and the perpendicular 4a, find 

each of the eight trigonometrical functions. 

16. Given 2 sin x = 3 cos y, and tan y = 3 cos x y to find sin #. 

[We have from equation (1) tan y = -, which, by means of 

equation (3), becomes -, and this in the second of 

cos y 

the given equations gives 

1 — cos 2 y — 9 cos 2 x cos 2 y, 



20 TB1GONOMETBY. 

from which finding cos y and substituting in the first of the 
given equations, gives sin x = ± 0.89 + or ± 0.56 +.] 

17. Given sin x + cos y = 1, and sin x cos y = — 4-, find sin a; and sin y. 

18. Solve the simultaneous equations tan x cot y = 2, and sin x cos 

19. Find sin x from the equations a sin * = £> cos - y , and sin a x = cos 6 y. 

[From the first equation, sin x log a — cos y log &; fiom the 

second, cos y = (sin .r) * Substituting gives 

log & , . v i . |~ . log 6 , . v i - 1 H ~ 

sm x — 1 ~ 5 — (sm x) h , or sm a; 1 - ,-^— (sm x) 6 = ; 

log a v L log a v J 

. *. sin x = 0, or sin x — ( te j 1 ) a ~ b • 

20. Find ?/ from the equations (tan a?)** - y = 5, (tan a:)' 203 f = £. 

21. Find the values of # from the equation 

sin a; cos \x tan Jos = 0. 

[Each factor may be zero, hence sin x = 0, and x — 0, #, 2tt, &c. 
If cos Ja; = 0, then \x = \it, %u, &c, and x = 7t, 3tt, 5^, &c 
If tan \x = 0, then Ja; = 0, ar, 2tt, &c, and a; = 0, 4?r, 8ar, &c] 



22. Functions of negative angles. If the angle be 
negative, and numerically less than 90°, it will be below 
the line OX in the figure on page 21, and the ordinate 
o will be negative, or — o, while the abscissa will be 
positive, or + a ; and we have 



Similarly, 



sin (— x) = = = — sin x. (11) 



cos (— x) = - = cos x. (12^ 

r 



FUNDAMENTAL PRINCIPLES. 



21 



It may be shown that the same relation exists when 
the angle exceeds 90° ; hence we have generally, 



cos ( — x) = COS X, 
sin ( — x) = — sin x, 

esc ( — x) — — CSC X, 



sec ( — x) = sec x, (13) 
tan ( — x) = — tan x, 
cot ( — sc) = — cot x. 



(14) 



That is, The cosine and secant of a negative angle are 
identical with that of a positive angle of the same magni- 
tude; but the sine, cosecant, tangent, and cotangent of a nega- 
tive angle are minus the like-named function of an equal 
positive angle. 

23. The complement of an angle is " 90° minus the 
angle." Thus the comple- 
ment of the angle X0P x 
will be 90 - X0P x =P 1 0Y; 
of X0P„ 90°- X0P 2 = 
— Y0P 2 ; and so on. 

The complement of an 
angle exceeding 90° will be ~~ x 
negative, and the comple- 
ment of a negative angle 
will be positive, and exceed 
90°. 

24. The supplement of an angle is " 180° minus the angle." 
Thus the supplement of X0P 2 will be 180°- X0P 2 = 
P,OX. 

25. Complementary functions. Let XOP — x, and POY 
= y ; then, because of alternate angles (Fig. on p. 22), 




x=X0P= OPY=90° -y. 



22 



TRIGONOMETRY. 



The angle x may be produced in this case, by revolving 
the generating line positively 90°, or to OT^jthen nega- 
tively through an angle y = YOP, giving x = XOP. 





From the figure and group (B) we have 

- = sin XOP = sin x = sin (90° - y\ 



and — == cos POY — cos y ; 

.-. sin XOP = cos POY) 





and this is true for any value of y ; and corresponding 
results may be found for the other functions ; hence we 
have for the functions of 90° — y> 



FUNDAMENTAL PRINCIPLES. 



23 



sin (90° —y) = cos y. 
tan (90° — y) = cot ?/. 
sec (90° — y) — esc y. 



cos (90° — ?/)=sin y. 
cot (90° — y)=tsmy. 
esc (90° — y)— sec ?/. 



(16) 



That is, TAe sine, tangent, or secant of an angle equals the 
co-function of its complement. 

[Cosine may be considered as an abbreviation for " the 
sine of the complement of an angle," and similarly for 
cotangent and cosecant.] 

26. Functions of 90° + y. 

Writing — y for y in the first of equations (16), the 
left member will become sin x = sin [90° — ( — y)] = 











sin [90° + y\ and the right member, cos (— y), which 
being reduced by the first of equations (14) becomes 
cos y ; and this is true whatever be the magnitude of y. 
Treating the other functions in the same manner, or 
by means of the figure, we have 



sin (90° + y) — cos y. 
tan(90°+y) = — coty. 
sec (90° + y) = — esc y. 



cos (90° +y) == — sin y. - 
cot(90° + y) = -tan y. 
esc (90° + y) = sec y. > 



en) 



24 TRIGONOMETRY. 

EXEECISES. 

1. What is the complement of 18°, 100°, 190°, - 40° ? 

2. What is the supplement of 15°, 120°, 200°, 300°, - 50° ? 

3. Construct the angle 90° + y f when y > 180° and < 270°. 

4. Construct the angle 90 ' + y, when y > 270° and < 360°. 

5. If sin x = cos nx, find x. 

[From (16) sin x = cos (90° — x\ which substituted in the ex- ' 

ample gives cos (90° — x) — cos nx ; . *. 90° — x = nx, and 

90° \ 

x— .1 

1 + n 1 

6. If cos x = sin 2x, find x. 

7. If tan x = cot no;, find a;. 

8. Find a; from the equation cot (90° — x) = tan (45° — a;). 

9. Given 2a; + 2/ = i# and sin x = cos 2y, find a; and y. 

10. Given sin a; = cos Sy and tan 3a; = cot y, to find a; and y. 

11. Given sin 3a; — cos 2y and 3a? + 2y = 90°, to find x and y. 

12. Prove that (a cos fif + (a sin /? sin yf + (a sin /i cos ^) 2 = a 2 . 

13. What angle is that of which the cotangent equals twice the cosine ? 

14. Show that if m 1 — ri 2 and 2mn be the lengths of the sides respectively 

of a right-angled triangle, the hypothenuse will be m 2 + ri 2 . 

[Substituting any pairs of numbers for m and n in the expres- 
sions in the preceding example will give respectively the sides 
and hypothenuse of a right triangle. Thus, if m = 3, and 
n — 2, the sides will be 5 and 12, and the hypothenuse 13.] 

15. Find the relations between the functions for an angle of 45°. 

[Substitute y — 45° in equations (16).] 

16. Eeduce equations (17) when y = 45°. 

17. Reduce equations (16) when y — 90°. 

18. Reduce equations (17) when y = 90°. 

19. Show that sin (90° — iy) = cos \y. 

20. tan (90° + \y) =? 

21. cos (90° -nij)--=? 

22. Given mx — ny — 90°, and sin ax — cos by, to find x and y. 

[From the first of (16) and the second equation of the example, 
we have 

cos by = sin (90° — by) = sin ax; 

.-. 90° - by -ax. 
Hence we find 

l + n -W,y= m ~? 90°.] 



an + bm an + bm 



FUNDAMENTAL PRINCIPLES. 



25 




27. Functions of 180° - y. In the first of equations 
(17), making y = 90° - y x , 
we have for the first mem- 
ber sin x = sin [90° + 90° 
-2/01 = sin (180° - yi ) ; 
and for the last member 
cos (90° — ?/i) = sin y l 
[from equations (16) ] ; 
hence, dropping sub- 
scripts, we have . 

sin (180° — y) = sin y. 

Or, from the figure, we have 

sin x = sin (180° — y) = ^4P -r- OP = o -r r = sin y 9 

and this relation is true whatever be the value of y. 

By means of either of these methods applied to deter- 
mining the other functions, we find 

sin (180° — y) — sin y. cos (180° — y) = — cos y. \ 
tan (180° -y) = - tan?/, cot (180° - y) = - cot y. I (18) 
sec (180° — y) = — sec y. esc (180° — y) = sec yl ) 

The first of these is, 27*e sme of an angle equals the sine 
of its supplement 

The second is, The tangent of an angle equals minus the 
tangent of its supplement. 



EXERCISES. 

1. Construct the angle 180 c - y, when y > 180° and < 270° 

2. Construct the angle 180' — y. when y > 270° and < 360° 

3. If sin ny' 2 = sin (180 c - y), find y. 

4. If cos (90 3 + ny) = cos (180° — my\ find y. 



26 



TRIGONOMETRY. 



5. Given tan (90° + ay) = cot (180° - by), to find y. 

[In this example y — 0, or a — b.] 

6. Is the relation tan (180° — y) = cot (90 D — y) possible? 

7. sin (180° -1?/) = ? 

8. cot(180°-iy) = ? 



28. Functions of (180° + y). 




— X- 




Substituting — ?/ for y in equations (18), or from the 
figure, we find 

sin (180° + y) = — sin y. cos (180° + y) — — cos#. j 
tan (180° + y) = tan y. cot (180° 4- y) = cot y. V (19) 
sec (180° + y) = — sec y. esc (180° + y) =• — esc y. ) 

29. Functions of (270° ± y). 

Making y == 90° — ?/ in equations (19) and reducing 
by means of equations (16) and (17), we have 

sin (270° ± y) = — cos y. cos (270° ± y) = ± sin y. ) 
tan (270° ± y) = zp coty. cot (270° ± y) = =F tany. [• (20) 
sec (270° ± y) = ± esc y. esc (270° ± y) == — sec y. ) 

The functions of % . 360° ± y are the same as for ± y. 
It will be noticed that the functions of 90° ± y, 180° ± y. 
270° ± y, may be expressed as functions of the angle y. 



FUNDAMENTAL PRINCIPLES. 



27 



EXERCISES. 

1. Construct 270° ± y when y > and < 90°. 

2. Construct 270° ± y when y > 90 < 180°. 

3. Find the functions of 270 = ± y when y — 45°. 

4. Construct 180° + y when y = — 30° ; also if y 

5. Show that sin 2 (270° ± y) + cos 2 (270° ± y) = 1 

6. Deduce equations (19) directly from the figure. 

7. Deduce equations (20) directly from a figure. 



= - 135°. 



TRIGONOMETRICAL LINES. 



30. Making r unity in group (B)> all the trigono- 
metrical functions may be represented directly by right 
lines. With a radius OA = OP = 1, describe a circum- 





ference, and AOP being the angle, draw EP and ^4T 
perpendicular to OA, and PT and (70 parallel to 
OA, then 



, ~ 7-v .EP P7P 



EP. 



An-o OE 

COS ^4 OP = jyp : 



OP 



= OE = FP. 



28 



TBIGONOMETR Y. 



AT AT 

tan AOP = ^4 = t=- = AT - 
OA 1 



cot A OP 
sec ^OP 



esc A OP = „ ~ 

vrs AOP = EA. 
cvsAOP= FC. 



OA 
AT 


CG 
0G~ 


CG 
1 


OT 
OA 


OT 
1 


: OT. 


_0G '■_ 


OG. 





CG. 





These results stated as follows are geometrical defini- 
tions of the trigonometrical functions. 

Let A be the origin of the arc, OA the initial radius, 
OP the terminal side of the angle, C, at a quadrant's 
distance from A, a secondary origin, and CD a second- 
ary diameter ; then in all the figures : 

The sine of an arc (or angle) is the perpendicular from 
the initial diameter to the terminus of tJie arc. 



FUNDAMENTAL PRINCIPLES. 29 

Thus, UP is the sine of the arc AP, positive above 
BA and negative below. 

The cosine of an arc (or angle) is the perpendicular from 
the secondary diameter to the terminus of the arc. 

Thus, FP = OE is the cosine of the arc AP, positive 
to the right of CD, and negative to the left. 

The . trigonometric tangent of an arc (or angle) is that 
part of the geometric tangent at the origin of the arc limited 
by the diameter prolonged through the terminus of the arc. 

Thus, A T, drawn tangent to the arc at A, and limited 
by the diameter through P prolonged, is the trigono- 
metrical tangent of A OP ; positive above OA, and nega- 
tive below. 

The cotangent of an arc (or angle) is that part of the geo- 
metric tangent at the secondary origin limited by the diameter 
through the terminus of the arc. 

Thus, CG drawn tangent to the arc at C, and limited 
by the diameter through P prolonged, is the cotangent 
of AOP; positive to the right of OC and negative to 
the left. 

The secant of an arc (or angle) is the distance from the 
centre of the arc to the extremity of the tangent. 

Thus, T, the distance from the centre to the ex- 
tremity T of the tangent A T, is the secant of A OP ; 
positive when the terminus P of the arc is on the secant, 
negative when P is on the extension of the secant, as in 
the second and third figures. 

The cosecant of an arc (or angle) is the distance from the 
centre of the arc to the extremity of the cotangent 

Thus, G , the distance from the centre to the ex- 
tremity G of the cotangent CG, is the cosecant of AOP ; 



30 



TRIGONOMETRY. 



positive when P is on the cosecant, and negative when 
on its extension. 

The versed sine of an arc (or angle) is the distance from 
the foot of the sine to the origin of the arc. 

Thus, EA is the versed sine of the arc AP, and is 
positive for all real arcs. 

The cover sed sine of an arc (or angle) is the distance from 
the foot of the cosine to the secondary origin. 

Thus, FC is the coversed sine of the arc AP, and is 
positive for all real arcs. 

It should be remembered that these lines, in trigono- 
metry, represent ratios — abstract numbers — and not 
linear quantities. 

31. The sine of an arc equals one- 
half the chord of double the arc. 

Thus, PB is the sine of the arc 
AP to the radius unity. But PB 
is one-half the chord PC of the 
arc PAC, which is twice the arc 
PA. 




EXERCISES. 

1. Find from a geometrical figure the limits of the trigonometrical 

functions, the values for which have been given on page 16. 

2. Construct an angle whose cosine is f. 

[In the first figure of article 30 take OE—\ OA, and erect the 
perpendicular EP ; the radius OP will be the terminus of the 
required angle]. 

3. Construct the cosine of an angle whose cotangent is 2. 

4. Construct all the trigonometrical functions of an angle whose 

tangent is — 1 and whose sine is negative. 



FUNDAMENTAL PRINCIPLES. 31 

. .1 i «_ * sin £ 

5. Show from a figure by means of similar triangles that tan x = -— j. 

LOS »£ 



Show that the following are true: 


6. 


sin 405° 


— sin 45°. 


7. 


sin 580° 


= — sin 40°. 


8. 


sin 45° ; 


> i sin 90°. 


9. 


(sin 60° 


+ sin 45°) > 1. 


10. 


sin (60° 


+ 45°) < 1. 


11. 


sin 90° 
cos 270° 


= oo . 


12. 


sin 180° 
cos 90° 


is indeterminate. 


13. 


sin 360° 
cosl80 J 


= 0. 



32. If x = sin y, we may write as its equivalent 
y = sin -) x* and read it "y equals the arc whose sine is 
x" or "y equals the angle whose sine is x" t 

The several expressions 

sin -) x, tan _) x, cot -) x, cos _) x, &c, 



* The inverse functions are usually written thus: sin- 1 x, where 1 is 
used instead of the curve as in the text. This form is supposed to have 
been suggested by algebraic expressions like y = m- 1 x ; but m _1 is a 
reciprocal, whereas the inverse functions are not reciprocals. The old 
notation is unfortunate, and has rather been tolerated than approved; 
and although it is generally better to continue the use of an objection- 
able notation, especially when in common use, than to introduce a new 
one, yet, in this case, the modification proposed is so slight and the 
form is so suggestive of the real nature of the function, we trust it will 
be approved. The suggestion of its use was made to the author by * 
Professor J. R. Paddock, of the Stevens High School. 

f The former reading is sometimes preferred because the terms of 
the equation are at once strictly homogeneous as line functions. Thus, 
y being an arc, is linear, and so also is the sine as a geometrical line. 
The latter, however, is equally correct. 



32 



TRIGONOMETRY. 




are called inverse functions, circular functions, or anti- 
functions. 

We will have 

sin sin -) x = x, sin -) sin y ~ y (21) 

J 3 . 

sin cos _) z = x = VI — ^ 2 . (22) 

sin -) cos 2/ = sin _) z — 90° — y. (23) 

Or, more generally, 
sin _) sin y = nit + (— l) n ?/, (24) 

where ?i is an integer, giving n values. But 

sin sin _) x 
has but one value, and equals x. 

33. The reciprocal may be indicated in the usual way, 
by a negative exponent, when no ambiguity results ; 
otherwise a parenthesis should be employed. 

Thus, 

1 



y = sin — = sin x _1 ; 

x 



y = sm~ > -= sin- } x 1 ; 

v x 



y 



cos y 



(cos 2/) -1 = cos 1 y ; 



(25) 
(26) 
(27) 



but the last form should not be used, for by most 
writers it designates an inverse function. 



y = — — : — = (sin _) x) 1 = sin 



-)-i 



x % 



FUNDAMENTAL PRINCIPLES. 33 

but the last form had better not be used for the reason 
just given. 

y = tan" } — — = tan~~^ (cot y~ x )~\ 
cot — 

y 



EXERCISES. 

1. If y = sin-> |, find cos y, and constrnct a figure showing the 

several parts. 

[Taking the sine of both members we have sin y — sin sin~> | = f ; 

. *. cos y = Vl — sin 2 y = ^ \ 5.] 

2. lfy = tan - ) 3, find sec y. 

3. If sin y = sin - ) a*, find z in terms of y and ?/ in terms of x. 

A n s. x — sin sin y, y — sin - ) sin - ) x. 

[sin sin y is not sin 2 y, but is the sine of an arc whose length 

equals sin y. Thus sin \n — sin 30° = |, and sin -J is the sine 

180° 

of an arc whose length is one half the radius, or \ x = 29| 

it 

degrees nearly.] 

4. By means of a figure or by analysis show that sin - ) x — csc~> — 




x 
5. Show that sin tan - ) x — 



VI + & 



6. If tan cos~> x = sec y, show that x = cos tan~) sec y. 

7. If sin cos - ) i = cos #, find sin x. 

8. If sin 2 y* — cos 0°, show that y —\7t' 

9. If sin~) y — cos~> a?, find ?/. ^ras. y = Vl — x* t 
10. If sin cos-) tan y = 0, show that y = {a. 

3 



34 TRIGONOMETRY. 

11. Show that sin 30° + sin 45° > sin (30° + 45°). 

12. Show that sin 45° + sin 45° > sin 90°. 

13. Show that 2 sin 30° > sin 2 x 30°. 

14. Show that cos 30° < 2 cos 60°. 

15. Find the trigonometrical functions of 135°. 

[Observe that 135° — 90° + 45% and substitute in equations (17). 
Or, 135 D = 180° — 45, and equations (18) will give the desired 
solution.] 

16. Find the trigonometrical functions of 120°. 



17. Given sec x — 



4 tan sin _) \ 
3 tan-e^r 



to find x. 



18. From a geometrical figure similar to those on p. 27, show that the 

sine of an angle equals the sine of its supplement, or sin 
(180° - x) - sin x. 

19. Show by means of the figures on page 28, that equations (19) and 

(20) are true. 

20. Prove that sin (y — 90°) = — cos y. 

21. Prove that cos (y — 90 ) = sin y % 

22. sin (y- 180°)=? 

23. cos(i/-180 c ) = ? 



34. Multiple angles. If x be an angle, nx will be 
an angle n times #, and is called a multiple angle. The 
general properties of multiple angles will be discussed 
further on, but the functions of certain multiples of \n 
may be readily found in this place. 

Tt 

Thus sin (hi f 1) ^ = 1, for all integral values of n ; 

for the angle will be (2n7t + ±tt) where 2n7r will be n 
complete circumferences, and \it added will make n cir- 
cumferences and 90 ° more ; hence the sine will be the 
same as sin 90°. 

Generally, if n be an integer, either positive or nega- 
tive, we have 



FUNDAMENTAL PRINCIPLES. 



35 



sin 

cos 



2nJ + y 



sin 

cos 



(2* + 1)1*-* 






(28) 
(29) 



-sin ?/ -j 
__cos 2/ J 

COS?/ - 

— sin y_ 

Functions which repeat themselves at regular inter- 
vals are called periodic functions. Trigonometry maybe 
defined as the algebra of periodic functions. 



EXERCISES. 



If n be an integer, including 0, show that : 

1. sin 4n— = ; sin {An + 2) '—— 0. 

2. sin (4» + 3) £•= - 1 ; cos {\.n + 2) \% = — 1. 

3. tan \ (2n7t) = 0; cos (4n + l)£ar ;= 0. 

4. tan (2» + 1) ^7r = ± oo ; cot \ (2n7t) = q= oo. 

5. sin (2ti + l)i^ = ( + , + ,-.-) 1 V2~ 



35. Impossible values. The rules applicable to real 
functions may be extended to operations upon unreal 
or imaginary functions. Thus, sin x = 2 is impossible 
according to the definition of the sine, but considering 
the equation to be true in fact as it is inform, we have 



cos x = Vl — sin 2 x= Vl — 4 = ' / — 3= a/3 V — 1 

sin a? 2 7 - , T 

tan a? = = — ~ V 3 v — 1, 

cos # o 

where the imaginary form shows that the original as- 
sumption was untrue. 



36 



TRIGONOMETRY. 



EXERCISES. 

1. If see x = i, find expressions for all the other trigonometrical func- 

tions. 

2. If tan x = V — 1, find expressions for the other trigonometrical func- 

tions. 

3. If cos x = 2, find the other trigonometrical functions. 

4. If Vsn x = — 1, find the other functions. 

5. Find tan x, given sec x = \ cos x. 

6. Find cos x if sin x = 2 tan x. 



36. The sinusoid. If on the right line AX, there be 
laid off AC = the arc Aa, Ae == the quadrant AB, and so 




on, and at C a perpendicular Cd — ab — the sine of 
aOA; at e, the perpendicular ef '= BO = sin 90°, and so 
on, and a continuous curve be traced through the 
ends of the perpendiculars, the curve is called " the 
sinusoid." The coordinates of its points are expressed 
by the equation y ~ sin x, and shows the relation be- 
tween an arc and its sine. Ag equals a semi-circum- 
ference ; and the sinusoid will cross the line AX at 
distances equal to multiples of a semi-circumference. 

All the other trigonometrical functions may be repre- 
sented in a similar way. 



EXEECISES. 

1. Construct a curve representing y — cos x. 

2. Construct y — tan x. 

3. Construct y — sec x. 

4. Construct y — vsn x. 



CHAPTER H. 



FUNCTIONS OF TWO ANGLES. 

37. To find the sine of the sum and sine of the difference 
of two angles. 

Let AOB and BOP be two adjacent angles, then will 
their sum be AOP. From \ p 

any point P in the terminal 
line OP, drop the perpendic- 
ular PC upon the initial line 
OA ; also PB upon OB, BA 
upon OA, and BD upon PC. 
Since PB and PB are per- 
pendicular respectively to the 
sides OB and OA of the angle 
AOB, the angle BPD will equal A OB. 

Let x = AOB = DP5, y = POP; then from the 
figure and the first of group (P) we have, 

. , ^ ■ OP CD + DP AB DP 

*™(x + y)=n»-- ~-or— = op+ of 




Bat, 
and 



OP ' 

AB AB OB 

m>=oB-op = * 1Jlxco *y> 

DP_DPBP 

Qp — pp ' Qp — cos x Sln 2/ 5 



therefore, sin (as + y) — sin as cos y + cos a; sin y. (30) 



38 TRIGONOMETRY, 

"Writing — y f or y in this equation, gives 

sin (x — y) = sin x cos (— y) + cos x sin (— y) 

= sin x cos ?/ — cos x sin ?/. (31) 

Equations (30) and (31) show that — The sine of the 
SUM of two angles equals the sine of the first into the cosine 
of the second plus the cosine of the first into the sine of the 
second. 

The sine of the diffekence of two angles equals the 
sine of the first into the cosine of the second minus the 
cosine of the first into the cosine of the second. 

38. To find the cosine of the sum and difference of two 
angles. 

From the preceding figure and group (B), we have, 

, .- v_ 00 _ OA-CA__ OA _DB 
cos {x + y) - Qp - - -gp Qp ^ 

OA OB_DB PB 
~ OB OP PB ' OP 

— cos x cos y — sin x sin y. (32) 

"Writing — y for y, gives 

cos (x — y) -- cos x cos y + sin x sin y ; (33) 



that is— The cosine of the DIFF S ^ NCE of tico angles equals 
the product of their cosines *^ s s the product of their sines. 



FUNCTIONS OF TWO ANGLES. 



39 



EXEKCISES. 

1. Represent geometrically the sine and cosine of x and y, and ofx — y 

in a single figure. 

2. Deduce equation (31) from a figure. 

3. Deduce equation (33) from a figure. 

4. Deduce equation (32) from (30) by 

writing 90° — x for x and — y 
for y. 

5. Develop sin (a + b + c). 

[Let a + b — x, then develop 
sin (# + c), after which sub- 
stitute the value of x and 
continue the development, 
finding 

sin (a + b + c) = sin « cos 5 cos c + cos a sin 5 cos c 

+ cos a cos 6 sin c — sin a sin b sin c. (34) 

6. Show that 




cos (# + y + £) = cos x cos y cos z — sin # sin y cos 
— sin x cos y sin z — cos a; sin y sin 0. 



(35) 



7. In exercise 6, make s = and reduce. 

8. In exercise 6 make z negative and reduce. 

9. In exercises 5 and 6 make x=y=z=a=b— c, and find sin 3x 

and cos 3x. 

10. Find from a figure sin (x + y) when x > 90° < 180°, and y > 

< 90°, and a + y < 180°. 

11. Find cos (x + y) when £ > 90° 

- < 180°, y < 90°, and x + y 
> 180°. 

12. Develop sin (Sx — 2y), after 

which make x — y. 

13. Given sin 45° = cos 45° — \ ¥2, 

and sin 30° = |, to find sine 

and cosine of 15°. 

[From equation (3), cos 30° = Vl — i = -J- VW; and these several 

values in (31) and (33), will give sin 15° = J ( V6— V2), and 

cos 15° = { ( VW+ V2).] 

14. Find sine and cosine of 75°. 




40 TRIGONOMETRY. 

[By means of (30) and (32) ; or by observing that 75° is the com- 
plement of 15°, we have sin 75 c = cos 15 c , and cos 75° = sin 15°.] 

5. Show that sin (45° + y) = i V2 (sin y + cos y). 

6. Find the sine of 22£°. 

[Let x = y = 22^° in (32), reducing by means of sin 2 22i° + cos 2 

22|° = 1, and find sin 22£° = i V%^- V2.] 

17. Find the sine and cosine of 7|°. 

18. Develop sin (90° - x). 

19. Develop sin 2 (x + y). 

20. Develop sin (x + yf. 

Prove the following statements : 

21. cos 75° = -^r 1 . 

2 V2 

22. sin (A + B) + sin ( A - B) = 2 sin A cos B. 

23. sin (A + B) - sin ( A - B) = 2 cos A sin B. 

24. cos (A + B) + cos ( A — B) = 2 cos J. cos J5. 

25. cos 4 9) — sin 4 cp — cos 2cp. 

26. cos (cp — 45°) + sin (^ — 45°) = iSTsin (p. 

27. cos (<p — 45°) = sin {cp + 45°). 

28. sin ncp cos cp + cos w<p sin cp = sin (w + 1) <p. 

29. cos (n — 1) cp . cos <p — sin (n — 1) q> . sin (p = cos ncp. 

30. sin w<p cos (?i — 1) cp — cos wcp . sin (n — 1) cp = sin <p. 

31. sin 60° + sin 30° = 2 sin 45= cos 15°. 

[Make 60° = 45° + 15°, and 30° = 45° — 15°, and develop.} 

32. sin Sep + sin 5q> = 2 sin 4<p cos cp. 

33. cos %7t — cos \ti — 2 sin -£ 2 -7r . sin -^tt. 

34. sin 2cp = 2 sin (p cos (p. 

35. cos (60° + cp) 4- cos (60° — <p) = cos cp. 

36. sin (fl + cp) . cos — cos (0 + cp) sin = sin cp. 

37. sin cp = 2 sin -££> cos £<p. 

38. sin \cp = 2 sin J<p cos •£<£>. 

3D. cos (a + /?) — sin (a — /?) = 2 sin (45° — a) cos (45° — fi). 
40. Solve the equation a sin <p + b cos cp= c. 

[Let m and /? be two auxiliary quantities determined by the rela- 
tion 

m sin fj = a. 
m cos fj — I ; 

from which we find tan fS = a + b, and ra = Va* + & 2 . 



FUNCTIONS OF TWO AXGLES. 41 

Substituting in the given equation gives 

m sin p sin cp 4- m cos p cos cp — c, 
or 

772 cos (/? — <p) = c ; 



c 



/. cos (p - cp) = - = „ 

w 1 a- + 6^ 

and 

cp = p- cos-) .1 

ya* - 6- J 

41. Find r and <p from the equations 

r cos ((p + a) = a, 

r sin (<p - p) = 5. 

[Expand, multiply the first by sin /J, the second by cos a, take 
the difference and find r sin cp. Then multiply the first by 
cos /i, the second by sin a, add and find r cos (p.] 

& cos « — a sin /> 

J.71S. <p = tan~) j—. -; 

sin a - a cos p 

b cos a: — a sin /3 



42. Given 



fZ cos a — a sm /i 

cos (a- — /j) sm tan~> =— -. ■ - a 

v ' b sm a + a cos /i 



r cos 9? cos 5 = — 2, 
r cos cp sin = +3, 
r sin /> = — 4, 



to find r, cp, 6. 



39. Discussion of equations (30), (31), (32), and (33). 
Let a? = y in (31) and (33), than 

sin 0' = sin as cos as — cos x sin as = 0. 

cos 0° = cos x cos x 4- sin x sin # = cos 2 x -f sin 2 a?. 



42 TRIGONOMETRY. 

But since x may be any angle, and cos 0° has but one 
value, it follows that sin 2 x + cos 2 x = a constant, and 
may be taken as unity. Hence, generally, 

sin 0° = 0, cos 0° = 1. 

Since 90° is the complement of 0°, we also have 

sin 90° = cos 0° = 1, 
cos 90° = sin 0° = 0. 

Letting x — y = 90° in (30) and (32), we have 

sin 180' = sin 90° cos 90° + cos 90° sin 90° 
= 1x0 + 0x1 = 0, 

and 

cos 180° = cos 90° cos 90° - sin 90° sin 90° = -1. 

Let x = 90°, then 

sin (90° +y) = sin 90° cos y + cos 90° sin y = cos y> 
sin (90° — y) = sin 90° cos y — cos 90° sin y = cos y, 
cos (90° + y) == cos 90° cos y — sin 90° sin y — — sin y 9 
cos (90° — y) = cos 90° cos y -f sin 90° sin y = sin y; 

the first and third of which have been previously de- 
duced, equations (17), and the second and fourth are 
in equations (16). 

Similarly, if x = 180°, we find the functions of the 
sine and cosine of 180° -f x and 180° — x as given in 
equations (18) and (19). 

Similarly, the functions for x == 270°, 360°, n~ 4m,7T, 

and any other angles may be found by substituting in 
the same equations. 



FUNCTIONS OF TWO ANGLES, 43 

Let x = 0, then (31) gives 
sin (— y) — sin 0° cos y — cos 0° sin y — — sin y ; 

and (33) gives 

cos (— y) — cos 0° cos y + sin 0° sin y = cos y; 

which are in equations (13) and (14). 

Equation (30) may be considered as a general equation 
in this science in the sense that when combined with the 
definitions of group (B) all other equations of trigo- 
nometry may be deduced from it. 



40. To find the tangent of the sum of two angles, and also 
of their difference. 

Dividing equation (30) by (32) gives 

, N sin (x + y) sin x cos y 4- cos x sin y 

tan (x + y) = — -7 — -^ = * = r- * 

v iJJ * cos {x + y) cos x cos y — sin x sin y 

sin x cos y cos # sin y 

, cos # cos y cos x cos ?/ 

~ by cos x cos y, = . . * 

J ^ 1 _ sin # sin 1/ 

cos x cog y 

reducing by means of equations (1) and (2), gives 

, , N tan x + tan y /om 

**(* + V) = 1 _to axtBin p (36) 

that is, The tangent of the sum of two angles equals the sum 
of their tangents divided by 1 minus the product of their 
tangents. 

Making y negative and reducing by equations (14) gives 

. . tan x — tan y ( ~ . 

^ y* ~ 1 + tan x tan y ^ ' 



44 TRIGONOMETRY. 

41. To find the cotangent of the sum and difference of two 
angles. 

Taking the reciprocals of the two preceding equa- 
tions, we have according to equation (10) 

if \ __ c °t # cot y — 1 

^ ^ ~ " cot x + cot y ^ ' 

. , x cot a? cot y + 1 ,„_ 

cot (x-y) =. — 7 --l — (39) 

v v/ cot y — cot x v y 



FUNCTIONS OF DOUBLE THE ARC. 

42. Making a? = y in (30), (32), (36), and (38), we have 

sin 2x = 2 sin x cos x (40) 

cos 2x = cos 2 # — sin 2 x (41) 

, 2 tan x 

^^ l -tan's ^ 

cot 2x = CO / g ,~ * (43) 

2 cot a? v J 



FUNCTIONS OF THE ANGLE IN TERMS OF HALF ANGLE. 

43. For x substitute \x in (40), (41), (42), and (43), and 
we have 

sin x = 2 sin \x cos ix (44) 

cos# = cos 2 \x — sin 2 {x (45) 

= 1-2 sin 2 ix (46) 

=: 2 cos 2 ix-1 (47) 



FUNCTIONS OF TWO ANGLES. 45 

2tan^ 

tan X = 1 - taa'fr . (48) 

cot 2 jag — 1 /ylm 

cot # = — ^ — t-t (49) 

2 cot ice v / 



FUNCTIONS OF ONE-HALF AN ANGLE. 

44. From (46) we have 



sin 4x = Vi (1 — cos x) (50) 



from (47), 

(50) - (51), 

(51) + (50), 

(44) + (47), 
(44) + (46), 
reciprocal (55), 
reciprocal (54), 



cos \x — Vi (1 + cos x) (51) 



timfr=i/ * cosa? (52) 

y 1 + cos x 



cot ix = jk/\ + ™l I 



1 — COS x 



sm x 



tan ix — =— ■ (54) 

^ 1 + cos # v 7 



cot hx — z. (55) 

1 — cos x v 



1 — cos x rt . a . 

tan ix = ; ■ (56) 

sm# v y 



. - 1 + cos oj /K _, 

cot \x = -. ■ (57) 

since v 7 



46 TRIGONOMETRY. 

45. From (30), (31), (32), and (33) we obtain by addi- 
tion and subtraction, 

sin (x + y) + sin (x — y) = 2 sin x cos ?/ (58) 

sin (a? + ?/) — sin (x — y) = 2 cos as sin y (59) 

cos (# + y) -f cos (x — y) — 2 cos x cos ?/ (60) 

cos (x + y) — cos (# — ?/) = — 2 sin x sin y (61) 



Let x -{- y = s and x — y = d, then a? = 4- (s + c?), and 
y =z ± (s — d), which substituted above gives 

sin s + sin cZ — 2 sin | (5 + d) cos i (5 — d) (62) 

sin s — sin d = 2 cos ? (5 -f c?) sin i (s — d) (63) 

cos 5 4- cos d == 2 cos i (s + d) cos i (s — d) (64) 

cos 5 — cos d = — 2 sin -J- (s + d) sin i (s — d) (65) 

which forms are convenient for logarithmic computa- 
tion. 



(66) 



(62) - (63), 

sin s + sin d _ tan | (5 + cf) 
sin 5 — sin d ~~ tan j, (s — d) 

(62) - (64), 

sin s + sin cZ 

(63) + (64), 

si n s - s in d . ,. 

cos,+ cosd = tan ^ S -^ ( 68 ) 

(62) - (65), 

sin s -f sin d , , , _ ,^ N 

^ = - cot + (s - d) (69) 

cos s — cos d x / K J 



FUNCTIONS OF TWO ANGLES. 47 

Formula (66) gives, — The sum of the sines of two angles 
is to their difference as the tangent of one-half the sum of the 
angles is to the tangent of one-half their difference. 



EXEECISES. 

1. tan-) f + tan-) J = \it. 

[Take tangent of the left member according to equation (36), 
tan(i?r) = l.j 

2. tan~) \ + tan-) \ — \ti. 

3. tan-) i + tan-) § = \it. 

4. tan-) a + tan~) b = tan-) a 



5. sin x = 



ab' 
2 tan \x 



1 + tan 2 \x 



[Substitute, tan ix =^ s j^i 

cos ix J 



6. cos x = 



7. tan \x — 



1 + tan 2 \x' 
1 + sin x — cos x 



1 + sin x + cos a;' 



8. tan (45° ± y) = cot (45° T y) = CQS ^ ^ s iM 

cos y T sin y' 

9. sin mx = 2 sin (m — 1) x cos a; — sin (m — 2) x. 

[In (58) make x - (m — 1) y. Another value for sin ma; may be 
found by making x = (m — l)y in (59).] 

. \ sin 2x = 2 sin a: cos a; 

sin 3a; = 2 sin 2# cos a; — sin a; — 4 sin x cos 2 a; — sin x 

sin 4a; = 2 sin 3a; cos a; — sin 2a; = 8 sin x cos 3 x — 4 sin a; cos a?. 

10. cos ma; = 2 cos (m — 1) a; cos x — cos (m — 2) x. 

[Deduced from (60). Another value may be found from (61).] 
. \ cos 2a; = 2 cos 2 x — 1 = - 2 sin 2 x+1 

cos 3x = 2 cos 2a; cos a? — cos x =4 cos 3 a; — 3 cos x 
cos 4a; = 2 cos 3a; cos a; — cos 2a; = 8 cos 4 x — 8 cos 2 a; + 1. 



48 TBIGONOMETBY. 

+ H . 3 tan x — tan 3 x 

11. tan3a; = — - _ — . 

1 — 3 tan 2 x 

[Develop tan (x + 2a?) by eq. (36), or make x = y — z in exercise 
12.] 

. rt , N tan x + tan y + tan z — tan a? tan y tan 2? 

12. tan [x + y + z) = = z -— - — ^ .. 

1 — tan x tan ?/ — tan x tan 2 — tan y tan s 

13. Given sin x — e find cos x. 

[e = 2.718281824 +, the base of the Naperian system of loga- 
rithms.] 

14. Given 

cos x = ~(e x " / ~ 1 + e-*^) (a) 

- smx=~(e xy/ ~ T - e- xx/:=1 ) V~l (b) 

to find cos- x + sin 2 x — 1. 

[Equations (a) and (b) are known as Eider's formula, having 
been discovered by the celebrated mathematician Leonhard 
Euler. They are proved by means of higher analysis, but 
are placed here for convenience of reference, and to afford an 
exercise in the use of imaginaries. Since they are true for all 
values of x, they illustrate the fact that imaginaries may be so 
combined as to represent real numerical values. One of the 
most interesting of these simple relations is shown in the 
mystical combinations given in the next exercise.] 

15. In equation (a) of the preceding example, make x — \it and 

i — V — 1, and show that 

1 
= [idijyzi = y> = 4.8095 +. 

16. In De Moivre's formula, which is 

(cos q) + V — 1 sin cp) n — cos nq) + V — 1 sin n<p, 

substitute q> = \n. and n — 1, 2, 3, etc., and reduce, finding 
the n th powers of V — 1. 




FUNCTIONS OF TWO ANGLES. 49 

17. By means of equations (a) and (b), exercise 14, show that 

V — 1 tan x = = . 

18. Generalizing De Moivre's formula, exercise 16, gives 

(cos cp + V — 1 sin cp)" — cos n (2m + <p) + V — 1 sin n (2m + <p), (c) 

where r — 0, 1, 2, 3, etc., and n a fraction. By means of this 
equation the several roots of unity may be found. 
What are the three roots of unity ? Make cp = 0, fl = i, and 
r — 0, 1, 2, in eq. (c). 

4ns. 1, i (- 1 + V^~3), i (- 1 - V^Sj. 

19. If 4 + £ + (7= 180% we may find 

cos A + cos B + cos =1 + 4 sin £ A sin |i? sin ^ 
sin J. + sin i? + sin =4 cos \A cos 4i? cos £6' 
sin 2A + sin 2i? + sin 2(7 = 4 sin A sin I? sin G 
cos 24 + cos 2B + cos 2(7 = — 4 cos J cos B cos (7 — 1 
tan 4 + tan i? + tan G = tan 4 tan i? tan 6 T 
cot \A + cot i£ + cot \G - cot \A cot i£ cot -|(7 
cos \A + cos ^i? + cos iC= 4 cos (45° + 44) cos (45° + J i>) 
cos (45° + W) 

The following series are readily proved by means of higher 

mathematics. 

_. . X X 3 X b X 7 

jfc «** = —- + --- + 

„o • , , y* 1 • 3 y 5 1 . 3 . 5 f 

22. 8m-> y = y + *§ + g-j. | + 3-^^. ^ + 

23.sin^,(l- f j)(l- 3 -g 2 )(l-£ 



24. cosz= ^-j^ 

„ * . 1 1 1 1 1 

2 °- 2" = 1 + 2 ' 8 + 2 • I ■ 5 + etC - 
[By making y = 1 in 22d example.] 
4 






50 TRIGONOMETRY. 

nn it 1 1 1 1 

26 -4 =1 -3 + 5-7 + 9 +CtC - 



27 4 



7T 1/1\ 3 1/1\ 5 . 

+ [^-IG) 3+ KI) 5+et 4 



^_2 2 4 4 6 6 

2 ~ 1 * 3 " 3 " 5 * 5 * 7 

which is known as Willis' expression. 

29. -^ 2 = 1-+ - + - + - + - + . (Todhunter). 

1 1 4 2 4 2 6 2 

30. _*■ = ! + -+ — - + 3 5 67<8 + etc. 

^ , H nx* n (3n - 2) x* n [15 (w - 1) 2 + 1] a 8 

31. (cos &)» = !- jg- + — - — [4 - — — - — ^ = L - + ... , 



46. The following methods will afford some variety in deducing 

certain expressions. 

Let AB be the diameter of a 
circle whose radius is unity, 
BAC=x t BAD = y. Draw CD 
CB, and BD. Then, 

AG = 2 cos x, AD — 2 cos y y 
CB = 2 sin a?, DB = 2 sin y, 
CD = 2 sin (a> + y), 

since the sine of # + # is half the 
chord of the arc CD. The product 
of the diagonals of an inscribed 
quadrilateral equals the sum of 
the products of its opposite sides, therefore 

AB . CD=CB . AD + AC . DB 

or, 

2 x 2 sin (x + y) = 4 sin # cos ?/ + 4 cos # sin y 

which, dropping the common factor 4, gives equation (30). 




FUNCTIONS OF TWO ANGLES. 51 

Constructing a figure with the angles x and y both on the same side 
of the diameter, the theorem for sin (x — y) may be as quickly proved 
in a similar manner. q 

In the annexed figure let the arc AD, or jS 
angle ACD = x ; draw DB, DA, CE ± DA, f J^ 
CG iBD; then 




Bf 
ACE = ABD = ADF= ix, 

AE = sin \x, CE = DG = \DB = cos \x, 

DF — sin x, CF = cos x> BF — 1 + cos x, FA — 1 — cos x ; 

and 

_ BF _ 1 + c os x 

cosix- m - 2cosix ; 

reducing, 

2 cos 2 -Ja; = 1 + cos x 
which is equation (51). 

Again 

. . DF sin x 
sin -ia? = -^ = 



BD 2cosia' 
. •. sin a; = 2 sin }2 cos \x 



which is equation (44). 
Again, 



tannic —-- 



'BF 1 + cos x 
which is equation (51). 

Again, 

FA 1 — cos £ 

tan && = =r=, = — : 

* DF sin x 

which is equation (56). 

Multiplying together the last two equations gives 

, /l — cos x 

tan -kc = if z 

V 1 + cos x 

which is equation (52). 



52 

Again, 
reducing, 



TRIGONOMETRY. 



FA 1 — cos x 

sin ^X = ^r-7 = -~ — = — ; — 

2 DA 2 sin £$ 



sin 2 \x = i (1 — cos a;) 



which is equation (50). 

47. Now constructing another figure, describe a circle with C as a 
centre, take AD = y, J.i? = a; — J.6r ; draw 
FD parallel to the diameter AJ, BEG, and 
DH perpendicular to AJ y also, FB, BD, DG 
and FG. Then will the arc DB = x — y, the 
chord 




BD = 2 sin i (z — y), !)# = sin y 
D# = 2 sin i (x + y), CH= cos y 
BE — sin a-, CjE/ = cos x 

MG = GE + J07 = sin x + sin y 

then, since BGD is an angle in the circumference and measured by \BD y 

. -n^ lr . t - x JfZ> -#// COS?/- COS # 

sm DGM, or sin -J- (# — y) = ~^r^ = ^tt\ = o — : — r~7 — ; — v» 
J v * ' GD GD 2 sm i (a? 4- y) 

which gives equation (65). 

Again, 

MD cos y — cos x 



tan £ (a - y) = 



which is the reciprocal of (69). 
Again, 



Jf6r sin <b + sin y' 



tan DFG = 



tan 4- (a: -f y) = 



if£ _ EGjhME _ EGjVjIE 
FM~ LM+ FL ~ LM + LD 

sin a; + sin y 



which is equation (67). 

Again, 

tan i (a; ■ 
which is equation (68). 

Again, 



cos x + cos y 



_ BM _ sin £ — sin y 
~ FM ~~ cos x + cos y 



tan | (a + y) _ MG __ sin x + sin y 
tan i (x — y) ~ MB ~ sin a; — sin y* 
which is equation (66). 



FUNCTIONS OF TWO ANGLES. 53 

We have 

FGB = 90° - GFM= 90° -* ( * + y) ; 
.-. FB-2 sin (i^B) = 2 cos £ (a? + 2/). 

Then 

. _ ^^ . , / v BM si n ^ — sin y 

sin J5i^D, or sin h (x — y) = ^^ = 5 r-. fr 

*' i^ 2 cos £ (# + y) 

which gives equation (63). 



48. Construction of tables. Since the student is not 
expected to make trigonometrical tables, we will only 
state briefly how they may be made. To make a table 
of sines, find the length of arc for the required angle, 
the radius being unity, and substitute in the second 
member of example 20, page 49, and reduce. Thus, to 

find the sine of 10°, the arc will be x — ^-^ x 10 

= 0.1745329, which substituted in example 20, above 
referred to, and reduced will give sin 10° == 0.17365 +. 
In this, or in some other way, the sine of all angles 
differing by 1', and in some cases by 1", have 
been computed and entered in a table. Then cos x 
== Vl — sin 2 x gives the cosines, tan = sin -f- cos gives 
the tangents, and cot — 1-4- tan, the cotangents. A table 
containing these values, is called a table of natural 
trigonometrical functions. The term natural is used to 
distinguish them from logarithmic functions. 

A table of logarithmic trigonometrical functions con- 
sists of the logarithms of the functions increased by 
10. Thus, sin 10° = 0.17365, and log sin 10° = log 
0.17365 - 1239670. Adding 10, we have 9.239670, and 
this number is entered in the table as log sin 10°. The 
number 10 is added to avoid negative characteristics. 
For directions for using the tables, see description just 
preceding the tables used. 



CHAPTEE III. 

SOLUTION OF PLANE TRIANGLES. 

49. Right angled triangles. The formulas in group 
(A) are directly applicable to the solution of right 
triangles. The parts of a triangle are its sides and 
angles. 




EXAMPLES. 

a. Solutions by means of Natural Functions. 

1. In the right triangle BAG, if the angle B = 43° 25' 

18", and the hypothenuse 
BG — 235, find the remain- 
ing parts. 

To find G. Since the three 
angles of a triangle = 180°, 
and .4 = 90°, therefore B+ C 
= 90°, and C = 90° - B ; hence G is the comple- 
ment of B ; and C = 90°- 43° 25' 18"= 46° 34' 42 ''. 

To find BA. From group (A) we have 

_ side adjacent __ BA _ c m 
hypothenuse BG a 9 

By means of a table of natural cosines, we find 
cos 43° 25' = 0.72637. To find it for 18" more, 
we assume that the function varies directly as 



SOLUTION OF PLANE TRIANGLES. 55 

the angle for 1' — which, though not exact, is 
sufficiently so for ordinary practice. From the 
same table, cos 43 J 26' - 0.72617, hence, 0.72637 - 
0.72617 = 0.00020 is the decrease in the cosine 
for T = 60" ; hence for 18 ' the decrease will be 
£f of 0.00020 = 0.00006, and this subtracted from 
the cos 43° 25', gives 

cos 43° 25 / 18" = 0.72631 ; 
.-. BA = 235 x 0.72631 = 170.68 + 

To find the natural sine, the difference for the 
seconds must be added to the sine of the degrees 
and minutes of the smaller angle. 

To find AC From group (A) we have 

. D AC b 
* mB =BC = a ; 
.\ AC = a sin B = 235 x sin 43 n 25' 18" 
= 235 x 0.68736 = 161.5296. 

Collecting results, we have 

Ans. (7=46 34' 42" 
BA = 170.68 
.4(7=161.5296. 

[It is generally advisable to compute all the known parts 
from the given ones, so that if an error is made in determining 
one of the parts it will not affect the computation of the other 
parts.] 

Chech As a check upon the work some of the com- 
puted parts may be determined by involving some 
of the other computed parts. Thus, having found 
the angle C and the side b, we have 

c — a sin C ~b tan C. 



56 TRIGONOMETBY. 

Also 

a 2 = V + c\ 

2. In the right angled triangle BAC the angle B is 

33° 30' and the side AC is 52.21. Find the other 
parts. 

Ans. BC = 94.6 ; AB = 78.88 ; C = 56° 30'. 

3. In the right angled triangle BAC the hypothenuse 

BC is 127.9 and the side CA is 97.72, to find the 
other parts. 
Ans. B = 49° 49' 15" ; A4 = 82.51 ; C = 40° 10' 45". 

4. In the right angled triangle D.ST' 7 , given the side EF 

75, and the side BE 50.59, to find the other parts. 

5. Given one of the acute angles 44°, and the hypoth- 

enuse 405, to find the other parts. 

6. In a right angled triangle one of the acute angles is 

67° 0' 25" and the side opposite is 710, to find 
the other parts. 

b. Solution by Means of Logarithmic Functions. 

7. Given a = 672.3412, B = 35° 16 / 25", to find the other 

parts. 

To find b. We have from group (A), 

sin B = -; 
a 

.\ b = a sin B; 

.\ log b = log a + log sin B. 

Add the logarithmic sine of B as found from the 
table of logarithmic functions, to the logarithm 



SOLUTION OF PLANE TBIANGLES. 57 

of a, then will the number corresponding to the 
sum of the logarithms be the required number. 
The work may be arranged thus : 

a = 672.3412 log 2.827590 

B = 35° 16' 25" log sin 9.761538 

b = 388.264. log 2.589128 



• • 



Ten has been dropped from the characteristic 
because the tabular logarithmic sine is ten too 
large. Finally, 388.264 is the number corre- 
sponding to the logarithm 2.539128. 

To find c. We have cos B — -; 

ci 

.'. log c — log a + log cos B. 

a = 672.3412 log 2.827590 

B = 35° 16' 25" log cos 9.911905 

c = 548.902 log 2.739495 

To find C. We have G = 90 - 35° 16' 25" = 54° 
43' 35"- 

b = 388.264 
c = 548.902 
C -54° 43' 35" 

Check. To test the accuracy of the work, compute the same 
quantity by different formulas. 

Thus, we have c = a sin C ; 

. \ log c = log a + log sin G ; 
or 

tan B = - ; 



. \ log tan i? — log b — log c. 



58 TRIGONOMETRY. 

If the last formula be used, employing the computed values 
of b and c, and the value thus found for B agrees with that 
given in the example, it is very certain that not only are the 
values of b and c correct, but also the process of verification. 
They will be absolutely correct if there are no errors balancing 
each other. 

8. Given the hypothenuse 65.07 and one of the acute 

angles 39° 38 28", to find the other parts. 

Ans. 41.51; 50.11; 50° 21' 32". 

9. Given the hypothenuse 2195 and acute angle 27° 

38' 50", to find the other parts. 

10. Given the hypothenuse 365 and one side 86, to find 

the other parts. 

Ans. 13° 37' 41" ; 76° 22 19 " ; 354.724. 

11. Given the hypothenuse 0.897 and one of the sides 

0.00086, to find the other parts. 

Ans. 3' 17 .7; 89" 56' 42".3 ; 0.89538. 

[To find an angle less than 2° from the logarithmic table, see 

EXPLANATION OF THE TABLES.] 

12. Given the hypothenuse 672.3 and one of the sides 

548.9, to find the other parts. 

Ans. 35° 17'; 54° 43'; 388.2. 

13. In a right angled triangle given one of the sides 

7643.5 and the angle opposite 37° 18', to find the 
other parts. 

14. Given the two sides 1728 and 1575, to find the other 

parts. 

15. Given the two sides 246.32 and 380.07, to find the 

other parts. 

16. In the figure if M represents the moon and EO the 

earth, the radius of the earth EO = 3956.2 miles, 



SOLUTION OF PLANE TRIANGLES. 



59 



and the angle OME = 57', required the distance 
ME, being a right angle. 

ME = 238,614 miles. 




17. If EN is a tangent to the moon's disc at N, the angle 

NEM will be the moons apparent semi-diameter. 
The apparent diameter of the moon being 31 ' 20", 
and the distance from the earth as found in the last 
example, what is the diameter of the moon ? 

Arts. 21748 miles. 

18. A tower 150 feet high throws a shadow 75 feet long 

upon the horizontal plane of its base ; what is the 
angle of elevation of the sun ? 

19. A railroad track makes a vertical rise of 150 feet in 

a distance of 3,000 feet, by uniform grade. What 
is the angle of the grade ? 

20. A rope whose length is 109 feet is fastened to the 

top of a tower 60 feet high, the lower end being 
carried as far as possible from the foot of the 
tower and fastened to the ground, required the 
angle which it makes with the horizon. 



60 



TRIGONOMETRY. 



21. Given the earth's equatorial radius 3962.8 miles, 
and the sun's horizontal parallax 8".87, find the 
distance of the earth from the sun. 

[The sun's horizontal parallax is what the apparent radius 
of the earth would appear to be if viewed from the centre of 
the sun.] Am. 92,152,000 miles. 



OF OBLIQUE ANGLED PLANE TRIANGLES. 

50. A general relation. The ratio of any two sides of 
a plane triangle equals the ratio of the sines of the angles 
opposite. 

Let ABC be any plane 
triangle the angles of which 
are denoted by A, B, C, 
and the sides opposite by 
"B a, b, c, respectively. From 
any angle C let fall the per- 
pendicular p upon the opposite side, or that side pro- 
longed if necessary. 





A B D D A 

From the right triangle ADC we have 



sin A =~, 
o 



SOLUTION OF PLANE TRIANGLES. 61 

and from BDG 

a 
and dividing the former by the latter, cancelling p, we 
have 

sin A a 



sin B b* 
Similarly, 



(70) 



sin G c J 
which was to be proved. 



a (71) 



Generally, 



a 



sin A sin B sin G' 



(72) 



The common value of this ratio is sometimes called 
the modulus of the triangle. 



51. Case I. Given two angles and one side. 
Let b be the side, A and B the angles. 

Then 

C = 180"-{A + B), (73) 

and (71), 

sin .4 
a — c - — ~, (74; 

sin 6 v J 

and (72), 

c = b -. — ^ (75) 

sin B v ; 



62 



TRIGONOMETRY. 



EXAMPLES. 

1. In the triangle ABC, let A - 48° 3', C = 40° 14', and 
and AC = 376, to find the other parts, 

£ From (73) 

.£ = 91° 43' 

To find c, use (75), as follows : 

B = 91° 43 ar. co. log sin 0.000195 
C = 40° 14' log sin 9.810167 

b = 376 log 2.575188 




~B~~ "D .-. c = 243 log 2.385550 

(The log sin 91° 43' is the same as log cos 1° 43.) 
To find BO = a, use (74), and find a - 279.8. 

Ans. B = 91° 43', a = 279.8, c = 243. 

2. In the triangle ABC, let ^=10° 12', 5 = 96° 36', 

and BC — 5.55, to find the other parts. 

3. In the triangle ABC, let ^L = 81° 30' 10", B= 40° 30' 

44" and J.i? = 696, to find the other parts. 

4. In the triangle ABC, let A = 68° 4' 20", 5 = 56° 32', 

35", and ^C = 696.75, to find the other parts. 



52. Case II. Given two sides and angle opposite one of 
them. 

Let a, b, B, be the given parts. 

To find A. From (72) we have 



sin A = t sin B. 



(76) 



But since the angle A and its supplement have the 
same sine, there will generally be two angles found by 
(76) which wdll fulfil the conditions of- the problem, and 



SOLUTION OF PLANE TRIANGLES. 



63 




hence two triangles, BAG and BA'C, and the result is 
said to be ambiguous. The several conditions involved 
in this case are most readily discussed geometrically. 

Thus, to con- 
struct the tri- 
angle, at B 
draw the lines 
BO and BA 
making ABC 
= B. On BC 
take the distance a, and with C as a centre and radius 
CA — b describe an arc AA\ and if it cuts BA in two 

points on the same side of B 
there will be two triangles. 

If b > a there will be only 
one triangle. For if B is 
acute the triangle will be 
ABC, but if B is obtuse the 
triangle will be BA ' C Both 
triangles are not admissible for the same data. 

If b equals the perpendicular from C to BA, and B is 

C c 





acute, there will be one triangle. But if B be obtuse 
there will be no triangle. 

If b = a and B is acute, there will be one triangle, 
ABC But iib — a and B is obtuse, or right, there will 
be no triangle, for b and a will coincide. 



64 TRIGONOMETRY. 

If B is acute, right, or obtuse, and b is less than the 
perpendicular from C to BD, 
there will be no triangle. 

If there be two triangles, rep- 
resent the two angles found by 
A' and A!\ and the two sides by 
B c and c". 




EXAMPLES. 

Given a = 31.238, b = 49, and A = 32° 18', to find B, 
C y and c. • 

ZbjSrad B. E q. (76) 

a = 31.238 ar co log 8.505316 

b =4:9 log 1.690196 

A = 32° 18' log sin 9.727828 

... B = 56° 56 57", . log sin 9.923340 
J9" = 123 d 3 3" 

To find C and C\ 

C = 180° - (32° 18' + 56° 56' 57" ) = 90° 45' 3" 
0" = 180° - (32° 18' + 123° 3' 3'' ) = 24° 38' 57" 

To find c and c". 

C = 90° 45' 3" log sin 9.999996 

O" = 24° 38' 57" , log sin 9.620196 

A =32° 18' " ar co log sin 0.272172 0.272172 

a = 31.238 log 1.494693 1.494693 



d = 58.460 log 1.766862 

c" = 24.381 1.387062 

( B' = 56° 56" 57" ) ( B" = 123° 3' 3" 

Ans. < C = 90 c 45' 3" [ or ] tf" = 24° 38' 57" 
(c' =58.460 ) (c" =24.381 



SOLUTION OF PLANE TRIANGLES. 65 

2. In the triangle ABC, let 5(7=94.26, .4(7 = 126.2 

and A = 27° 50', to find the other parts. 
B= 38° 52' 46". 0= 113 3 17' 14". c = 185.439. 
or 141° 7' 14". 11° 2' 46". 39.682. 

3. Given a = 91.06, b = 77.04, ^ = 51° 9' 6". Find the 

other parts. 

B = 41° 13' ; = 87° 37' 54" ; c = 116.82. 

4. Given a = 1257.5, c = 1751, A = 31° 17' 19". Find 

the other parts. 

5. Given a = 40, b = 55, A = 60°. Arts. Impossible. 

6. Given a = 43, b = 45, A = 120°. ^4/is. Impossible. 




53. Case III. (yivew fotfo sides and the included angle. 
Let a, &, (7, be the given parts. 
From (70) we have 

a __ sin A 
b ~ sin B 

Adding one to both members, we A 
have by reduction, 

a + b _ sin A + sin 5 
6 sin J5 

Similarly, subtracting 1 from both members, 

a — b _ sin ^ — sin 5 
6 sin 1? 

Dividing gives, 

a -\- b _ sin ^4 + sin B 
a — b~ sin A — sin 5* 



66 TRIGONOMETRY. 

This reduced by (66) gives 

a-b~~ttm%(A-B)> Kii) 

that is, The sum of tico sides of a plane triangle is to their 
difference, as the tangent of half the sum of the angles op- 
posite is to the tangent of half their difference. 

To find A + B ice have 

A + B = 180° - G. (78) 

To find A — B we have from (77) and (78) 

°^-fitan(180°-(7) 
_a + b _ 



A-B = 2 tan" 



(79) 



Keducing the quantity in the brackets, [ ], and find- 
ing the angle corresponding to this tangent by means 
of a table of trigonometrical functions, and multiplying 
the result by 2 will give A — B. 

To find A and B. We have from (78) and (79) 

A = i [ (A + B) + (A <- B) ] (80) 

B = i[(A + B)-(A-B)] (81) 

To find c. We have from (71) 

Sil1 Q /DON 

c — - — -, a (82) 

A solution may also be made by means of right tri- 
angles, as will be shown in article 55. 



EXAMPLES. 



1. In a plane triangle ABC, given a — 748, 6 = 375, 
C= 63° 35' 30", to find the other parts of the 
triangle. 



SOLUTION OF PLANE TRIANGLES. 67 

a + b = 1123 a. c. log 6.949620 

a-b = 373 log 2.571709 

i(A + B) = 58° 12' 15'' log tan 10.207660 

.\ i(A — B) = 28° 10' 54" log tan 9.728989 
A = 86° 23' 9" 

B = 30° 1 21" a. c. log sin 0.300744 

(7^63° 35' 30" log sin 9.952137 

b = 375 log 2.574031 

c = 671.29 log 2.826912 

Ans. A = 86° 23' 9", B = 30° V 21", c = 671.29. 

2. Given c = 304, a = 280.3 5 = 100°, to find the other 

parts. 

Ans. A = 38° 3 33", C=- 41° 56' 57", & - 447.86. 

3. Given a = 468, b = 365, = 82° 22' 22', to find the 

other parts. 

4. Given a = .3756, 6 = .2375, C= 68° 25', to find the 

other parts. 



54. Case IV. Given the three sides of a triangle, to find 
the angles. 

c 





From the figure we have 

AD = x' = b cos A, DB = x" 



a cos B, 



68 TRIGONOMETRY. 

•\ x f + a?" — c = 6 cos J. + a cos B 
similarly, b = a cos + c cos J. ^ (83) 

a = c cos B -\- b cos (7 

In the second figure cos B is negative ; hence x" will 
be negative, and we have 

c — x' + {— x") = x — x" — b cos A — a (— cos 5) 
= 6 cos A + a cos i? 
as before. 

From the 1st of (83), 

a cos B= c — b cos ^4 
squaring, a 2 cos 2 B = c* — 2bc cos A + V cos 2 ^4,. 
from (72) a 2 sin 2 5 = V sin 2 J., 

adding, a 2 = c 2 + ft 2 — 2bc cos J. 

similarly, 6 2 = a 2 + c 2 — 2ac cos 5 )>- (84) 

c 2 = a 2 + 6 2 - 2ab cos (7 

From (84) we have 

- a 2 + & 2 + c 2 



(85) 



2a& 

Equations (85) will solve this case by means of 
natural functions. To adapt them to logarithmic com- 
putation, they may be transformed thus : 

Subtracting each member of the first of (85) from 
unity we have 

- A 2bc + a* -F - c 2 a 2 - (6 - cY 

_ (a + b — c) (a — b + c) 
~ " ~2~bV~ 



\J\Ji3 






2bc 




COS 


B = 


a 2 


-&' + 


c 2 




2ac 




COS 


(7 = 


a 2 


+ V- 


c 2 



SOLUTION OF PLANE TBI ANGLES. 



69 



But 1 - cos A = 2 sin 2 \A (eq. (50) ) ; and letting 2s 
= (a + b + c), we have a + 6 — c = 2 (* — c), and 
a — 6 + c = 2 (s — 6) ; hence 



sin 2 i 



_ (s - b) {s - c) m 
- be » 



similarly, 



sin 



ii _ j A* ~ h ) ( s - c ) 
2 ~~ Y be 



sin £ 



B ^ a/ {* -a) {?- c) i 






(86) 



Adding each member of (85) to unity, and reducing 
in a similar manner, we find 



cos 



6c 



i z> i A (s ~ b) 
cos |5 = r — 



ac 



cos 



^=^ } 



(87) 



Dividing each of equations (86) by the corresponding 
one of (87), gives 



tan \A = / (S ~ *> (S " C) 

s (s — a) 

s (s — b) 
taniC^^SEpSEl 

5 (5 — C) 



(88) 



70 TRIGONOMETRY. 

EXAMPLES. 

1. In a plane triangle, given a = 30, b — 25, and c = 20, 

to find tlie angles. 

By natural functions. The first of equations (85) 
gives 

' V + e - a 2 625 + 400 - 900 
cos A = Wc iooo = °- 125 ' 

hence 

A = 82° 49' 9". 

In a similar manner the values of B and C may be 
found. 

2. In a plane triangle a = 21.35, b = 12.17, c = 10.08, to 

find the angles. 

By logarithmic functions. Using equations (88), 

log log. log 

5 = 21.80 a. c. 8.661544 a. c. 8.661544 a. c. 8.661544 

s-a= 0.45 a. c. 10.346787 1.653213 1.653213 

s- 6= 9.63 0.983626 a. c. 9.016374 0.983626 

s - c = 11.72 1.068928 1.068928 a. c. 8.931072 

2)1.060885 



\A = 73° 34' 25" .... tan 0.530442 2)18.400059 

iB- 9° 0'22."5 tan 9.200029 2)2.229455 

\C= V 25' 12. "5 tan 9.114727 

Ans. A = 147° 8' 50" B = 18° 0' 45" t= 14° 50' 25". 

3. Given a = 14, 6 = 17, c = 19, to find the angles. 

4. Given a = 0.0237, b = 0.164, c = 0.0843, to find the 

angles. 



55. Solution by means of right triangles. Let ABC be 
any plane triangle, CD a perpendicular from any angle 



SOLUTION OF PLANE TEIANGLES. 



71 



C upon the opposite side AB. We may indicate briefly 
the solutions for the several cases. 





Case I. Given one side and two angles. 

Let AG — b, be the side, A and Cthe angles, then 

B = 180° - (A+ C); BCD = 90° -B; 

ACD = 90° -A, (89) 

AD — b cos A — x \ CD = b sin A = p ; 

CB= CD - cos BCD = a; (90) 

DB = a cos B = x' ; .-. AB=x' + x" ; (91) 

observing that #" in the second figure will be negative, 
since cos B will be negative. 

Case II. Given two sides and an angle opposite one of 
them. 

Let a and b be the sides and A the angle. Find AD 
and CD as in Case I. Then sin B = p -f- a ; (7 — 
180° — (^4 + jB) ; BD — a cos B\ hence all the parts will 
be found. 

Since B generally has two values in the equation 
sin B — p -f- a, there will generally be two solutions. 

Case III. Given tico sides and the included angle. 

Let the given parts be b, c, A. Find AD — x, 
and CD =p as in Case I. Then DB = c — x = x" ; 
tan B^p~x") DCB = 90° - B ; u4CD = 90 -^ 



72 TRIGONOMETRY. 

.-. ACB =90°- (A + B) 9 and all the parts become 
known. 

Case IV. Given the three sides. 

Let CD =p,AD = x, BD = x" ; then 



p* = f - 

and 


■x'*- 


= a 2 - x"\ 


x' 


+ x" 


= c, 


These equations give 






, - a 2 + V + c* 

X =z 




_„ a 2 - 6 2 + c 2 


2c 


* ~ 2c 


Then 






A X ' 

cos A = T 






cos B = — ; 
a 



(92) 



(93) 



in which if x or &'' be negative, the corresponding 
angle will be obtuse. The angle C may be found by 
dropping a perpendicular from A or B and proceeding 
as before ; or, simply, C = 180° — (A + B). 







EXAMPLES. 




1. 


Given a =250 


.4=10° 12 


5=46° 36' 




find (7=123° 12' 


c=1181.30 


6=1025.74 


2. 


Given a=17.432 


J=19 574 


^=38° 44' 12" 




find c'=27.908 


#=44° 1' 28" 


C"=97°44'30" 




c"=2.8404 


B"=1S5° 58' 32" 


<7"=5°47'16" 


3. 


Given a=95.98 


b= 66.28 


(7=175° 19' 10" 




find A=2° 46' 8" 


5=1° 54' 42" 


c=162.128 



4. Given a=6, 6=8, c=10; find the angles. 



SOLUTION OF PLANE TBIANGLES. 73 

5. Given a = 15, b — 15, c = 17 ; find the angles. 

6. Given a = v 711 ^ b=V — 3, c = V — 7; find the angles. 

7. Given a = 3 V^T, & = 4 V 3 !, J?= 60° 38', to find 

the other parts. 



56. Small angles. For tables which give the values 
of the trigonometrical functions to minutes only, the 
student is obliged to compute the values for the seconds. 
This is usually done by assuming that the function 
varies directly as the angle from minute to minute. 
But for angles less than two degrees and near 90°, 
this is not sufficiently exact ; hence these angles should 
be avoided, if possible, unless special tables are pro- 
vided, which either give the values of the functions 
to seconds, or which give the means of computing the 
values of the small angles more accurately. In this 
work, the latter plan is adopted, as may be seen by 
consulting the first two pages of the table of logarithmic 
trigonometrical functions. 

In this case the values of 



, sin x . _ _ 

log = log sin x — log x = q — I (94) 

tan nr 

log = log tan x — log x = q — l (95) 

are computed for each minute, and change slowly, as 
may be seen by examining the table, where even the 
sixth figure, in some cases, is the same for several 
minutes of arc. If the angle be given in seconds, we have 



74 TBIGONOMETR T. 

smx= x ; (96) 

.\ log sin x = log x" + q — l 9 (97) 

or log x" = log sin ^? — (g — Z) (98) 

Similarly, log tan x = log x' + (q — I) ; 

.\ log a?" = log tan x — (q — I), (99) 

where q — I in the latter case is taken from the column 
next to that for tangents. 



Example. Given the base of a right triangle 4500 and altitude 5, to 
find the angles. 
We have 

tan A = j=twx ; •*. log tan A — log 5 — Log 4500, or 

log 5 = 0.698970 
log 4500 = 3.653213 

difference, 3.045757 

adding 10 

gives 7.045757 

subtract q-l 4.685575 

.'. z" = 229".l log 2.360182, 

which is correct to 0.1 of a second. The other angle will be 89° 
56' 10".9. 

Having found 7.045757, we find the logarithm corresponding most 
nearly to it in the column having "Tang" at its head, and in the 
fourth column we find 4.685575 for the value of q — I, which being 
subtracted gives the logarithm of the seconds ; the number of which is 
found in the table of the logarithms of numbers. For further instruc- 
tion See EXPLANATION OF THE TABLES. 



AEEA OF TKIANGLES. 

57. The area of a triangle equals one-half the product of the base by 
the altitude, or 

K=icp (100) 



SOLUTION OF PLANE TRIANGLES. 



75 



where c = the base, p = the distance of the base from the opposite 
angle, and K = the area. 

a. Given two sides and the in- 
cluded angle, or b, c, A. 

Here 

p = b sin A, 
.-. K=ibcsmA. (101) JT~ c D~ "~ B 

b. Given two sides and an angle opposite one of them, or a, b, B. 




sin A z= — 



^sin.B; 



sin B 



Then 
p" = a sin C= a sin ( A + B) = c 



= sin->( 

in ^sin-) (J 



sin B ) + B 



Area = \ab sin 



sin-) ( t- sin B ) + B 



(102) 
(103) 

(104) 
(105) 



Expanding the sine of the quantity in the brackets by equation (30) 
and reducing, gives 

Area = $asinB [a cos B + Vb 2 - a* sin 2 B~\ . (106) 

Interpreting this result by means of the figure, we find that if a per- 
pendicular be dropped from C upon the side c, dividing the latter into 
two segments, the term a cos B in the brackets will be the length 
of one segment, and the other term, V& 2 — a? sin 2 B, will be the 
length of the other segment, and hence their sum will be the length of 
the unknown side c, and the factor a sin B will be the length of the 
perpendicular from O upon the unknown side c. 

If a, b, A, be given, the solution will be of the same form and the 
result may be found by writing A for B in equation (106) and inter- 
changing a and b. 

c. Given two angles and the included side, or A, B, c. 
First find a from equation (71), 

a _ sin A sin A 

c ~ 



sin G ~ sin ( A + B) ' 

sin A 
C sin (A 4- BY 



(107) 



76 



TRIGONOMETRY. 



then, P = a sin B 

sin J. sin 5 
substituting, = c s ^j~^B) ; 



(108) 



- Area - ic sin(A + R) 

d. Given the three sides, as a, 5, c. 
Let AD = a;, then DB = c — x, 

and ^ = *"-rf = a« -(«-#; (HO) 



_ ^ + & 2 + c 2 
■'•■= 2c ' 

which substituted above give8 

r _ a 2 + 6 2 + e*\ a 45 2 c 2 - (- a 2 -f V + c 2 )* 



(HI) 



**=**-( — a — ) = 



4c 2 



which being the difference of two squares reduces to 

, (25c - a 2 + b 2 + c 2 ) (25c + a 2 - b 2 - c 1 ) 

P = 4C 2 

_ [- a 2 4- {b + c) 2 ] [a 2 - (& - c) 2 ] 

"" 4c 2 

— (^ + ^ + C H— ^ + 5 + c) (a — 6 + c) (a + & — c) 
4c 1 ' 

__ 4s (s — a) (5 — 5) (5 — c) 



-•• P = ~^ 5 (5 - a) (s - b) (s - c) (112) 

c 

where s = | (a + b + c). The area will be 

K = Vs (s - a) (s -&)(*- e). (113) 

Or terms; CD = p — b sin J. 

Eq. (44), = 2b sin \A cos ^ 

Eqs. (86), (87), = - V* (s - a) (s - b) (s- c); 

c 



SOLUTION OF PLANE TRIANGLES. 77 



Eq. (100), . '. K = Vs {s - a) (s - b) {s - c), 

as before. 

If p' be the perpendicular from the angle A to the side a we may 
write, from inspection of (112) 



P' = -Vs{s- a) {s -b)(s-c); (114) 

Cb 



and similar for the perpendicular from B to &, 



and hence, 



p" = -Vs(s-a) {s -b){s-c)\ (115) 



L+^ + l * (116) 

p p p K 



58. Inscribed circle. If a circle be inscribed in a triangle, and the 
centre joined with each of the angles by a right line, it will be divided 
into three triangles each having an altitude equal to the radius of the 
circle. Hence the area of the triangle will be 

K= \ra + \rb + \rc = rs; (117) 

.-. r = ? = \ Ys{s-a)(s-b)(s-c) =g* (118) 

59. Circumscribed circle. If R be the radius of the circumscribed 
circle, we will have 

B= & = # = * (119) 

sin A sin B sin G 

60. Escribed circles. If r', r", r"', be the radii of the escribed cir- 
cles, opposite respectively to the sides a, b, c, then we have, 

r' = ~^-, r' = ^-r, r';-— (120) 

s — a s— b s — c 

1 1 1 1 

- = ZJ + j, + ~Jn (121) 

r r r r 
K 2 =:r .r' .r" . r'" (122) 

(Escribed circles are exterior to the triangle, tangent to one side and 
to the other two prolonged.) 



78 



TRIGONOMETRY. 



EXERCISES. 

Find the area of the triangles, the three perpendiculars, the radii 
of the inscribed, circumscribed, and escribed circles, given 

1. a =12.5, & = 15.2, c = 20.5. 

Ans. K= 94.641 + , R = 10.288+, r = 3.927+. 
# = 15.144+, jo' = 12.452+, p" = 9.223+. 
r'= 8.156 +, r" = 10.633+. r'" = 26.289+. 

2. a = 16.2, b = 22.2, C = 30° 20'. 

^Lns. JST= £0.814 +, 22 = 11.316+ r = 3.632+. 
# = 11.211+, #'=8.181+ #'" = 15.617+. 
r = 10.319 + , r" = 32.433 + r'" = 6.777 +. 

3. a = 12.5, 5 = 40° 40', (7 = 60° 10'. 

4. A = 100° 10' J5 = 50° 25' c = 20.5. 



PRACTICAL EXAMPLES. 

1. Find the area of a regular pentagon of which the sides 

are 12 feet each. 

2. Find the area of a regular hexagon of which each 

side is 12 feet. 



3 3. To determine the unmeas- 
ured distance AB, hav- 
ing given AC = 235 feet, 
CB = 317 feet, and the 
angle ACB= 48° 45' 12". 



4. From the top of a light- 
house 100 feet high, the angle of depression of 
a ship lying at anchor was 7° 43' 43 ", from the 





SOLUTION OF PLANE TRIANGLES. 79 

foot of the lighthouse the angle of depression 
was 6° 12' 13" ; required the height of the hill 
on which the light- 
house was situated v "~77 B 
and the horizon- 
tal distance to the 
ship. 

5. To determine the inac- 

cessible distance 
AB, having given 

CD = 225 feet, ACS = 74° 15', 
BCD = 44° 27', BDA = 66° 54', 
and CD A = 39° 38'. 

Ans. 528.265. 

6. A headland was observed from a ship to bear directly 

east ; after sailing S.W. 25 miles the same head- 
land bore E. by N. ; required the distance of 
the headland from the two points of observa- 
tion. 

[Instead of dividing the quadrant into 90 degrees, navigators 
divide it into eight equal parts called points, which are sub- 
divided into quarters. The points as named on the card of 
the mariner's compass in the quadrant between north and east 
are as follows : 

N., N by E., NNE, N.E. by N, N.E., NE. by E. 
E.N.E., E. by N. y E., and similarly for the other quadrants. 
Thus, N hy*E. will be 11^° E. of N; NNE, 22^° E. of 
N, and so on.] 

7. At one station a cloud bears N.N.W. and its angle 

of elevation is 53° 22' ; at a second station, whose 
bearing from the first is N. by E., and distance 
one mile, the cloud bears west ; required the 
height of the cloud. Ans. 1.427 m. 



80 



TRIGONOMETRY. 




8. How far from the foot of a light-house can a light 
350 feet high be seen out at sea, the radius of the 
earth being 3,956 miles ? Ans. 22.901 m. 

9. To determine the height of 
the pole FD, the base AB 
— 250 feet was measured, the 
end A of which was AG = 12 
feet higher than the end B. 
When the transit was at B, 
the point D, where the hori- 
zontal line BJD cut the pole 
was noted, the vertical angle 
DBF = 12° 24', and the hori- 
zontal angle DBO = 35° 15' were measured ; and 
at A the horizontal angle = DCB = 27° 51'. Ee- 
quired the height FD. As a check upon the 

c work, at A the point E may 

be noted, and the vertical 
angle EAF measured, by 
means of which FFmsij be 
determined, which added to 
JED should equal DF. 

10. From a point P were meas- 
ured the angles APG~ 
33° 45', and CPB = 22° 30'. 
The distances AG = 600, 
AB = 800, BO= 400, being 
known, it is required to 
find AP and BR 

Draw a circle through the points A, B, P ; join (7 and P, 
and from 2£, where CP intersects the circle, draw EA and 
EB ; then will EAR = CPB } and ABE= APC } since the 





SOLUTION OF PLANE TRIANGLES. 81 

former angles are subtended by the arc EB y and the latter 
bjAE. 

To find the point P by construction. 

On one side of iC as a chord describe a segment of a 
circle which will contain an inscribed angle = 33° 45', which 
may be done by making A CO = 90" 
— 33 J 45' = 56° 15 ', and intersect- 
ing CO by the perpendicular DO 
drawn through the middle of AC; 
the intersection will be the cen- 
tre of the arc, and OC the radius. 
Similarly, on the chord BC describe 
another segment of a circle that 
will contain an angle of 22° 30'. 
The intersection P, of the two arcs will be the point sought. 
But four such arcs may be described, two having their centres 
on opposite sides of AC, and two others having their centres 
on opposite sides of BC ; and the corresponding arcs will 
generally intersect in four points which may be designated 
by P,, P 2 , P 3 , P 4 . There can be no more than four points, 
but there may be less. In this problem there are four, as 
follows : 

( AP X = 710.2 ( APi = 910.3 ( ^4P 3 = 164.3 C AP, = 981.2 
< PP, = 934 3 •] BP, = 764.9 ] BP, = 960.5 < BP, = 185.5 
( CPi = 1042.5 ( CP, = 434.0 ( CP 3 = 729.6 ( CP, = 565.1 

11. From one station the angle of elevation of tlie top 
of a tower was observed to be 75° 18', from an- 
other station 400 feet farther from the tower the 
angle of elevation was 28° 45' ; required the 
height of the tower, the two stations being in the 
same horizontal plane. 

6 



82 SPHERICAL 

SPHERICAL TRIGONOMETRY. 

GEOMETRICAL PRINCIPLES. 

61. A spherical triangle is a three-sided figure on tlie 
surface of a sphere, each side being the arc of a circle. 
But since the sides of spherical triangles in ordinary 
practice are arcs of great circles, we will consider that 
— A spherical triangle is formed by the arcs of three great 
circles. 

A great circle is one whose plane embraces the centre 
of the sphere. All other circles are called small circles. 
A great circle divides the surface of the sphere into 
two equal parts ; two great circles will generally divide 
the surface into four parts ; and three great circles 
into eight parts, or eight triangles. 

The sides or angles of some of these eight triangles 
will exceed 180°, but in this treatise only those tri- 
angles will be considered in which each part is less 
than 180°. 

The centre 0, of the sphere being common to the 
planes of all the great circles, the intersections of these 
planes necessarily radiate from that point, forming a 

trihedral — A CB, having 
the three facial angles A OC, 
COB, A OB, and three dihe- 
dral angles, one at each of 
the edges of the trihedral. 
If with as a centre and a 
common radius OA, arcs AB, 
AC, BC, be described, they 
will be on the surface of a 
sphere, and measure the facial angles which they sub- 
tend. 




TRIGONOMETRY. 83 

The dihedral angle between the planes A OC and 
A OB will be the same as the angle at J. between the 
arcs AB and AC; for each will be measured by the 
angle between two lines, one in each plane, respectively 
perpendicular to the common intersection OA, drawn 
at A or at any other point of OA. 

Spherical trigonometry, which is the science of the 
spherical triangle, may also be defined as the science of 
the trihedral. 

It is proved in geometry, that in the spherical trian- 
gle which we are to consider : 

The sum of the three angles may be anything be- 
tween 180° and 540°. 

The sum of the three sides is between 0° and 360°. 

Each side is less than the sum of the other two. 

Each side is the supplement of the angle opposite in 
the polar triangle. 

Each angle is the supplement of the side opposite in 
the polar triangle. 

The pole of an arc is the point where the axis of its 
great circle pierces the spherical surface. The arc of a 
great circle may be described by revolving one end of a 
rigid quadrant of a great circle about any point as a 
pole ; the remote end will trace the required arc. A 
string may be used in place of the rigid arc. 



EXERCISES. 

1. Describe the arc of a great circle on a sphere. 

[If no sphere be at hand, one may be extemporized by using an 
apple, ball, or some object only approximately spherical, and the 
solutions indicated.] 

2. Describe two arcs of great circles mutually perpendicular. 



84 SPHERICAL TRIGONOMETRY. 

[The pole of the second arc will be at some point on the first 
arc] 

3. Construct three mutually perpendicular arcs of great circles. 

[They will form a tri-rectangular triangle.] 

4. Construct two mutually perpendicular arcs of great circles, and cut 

them by a third great circle perpendicular to one and oblique to 
the other. 

5. Construct two mutually perpendicular arcs of great circles, and cut 

them by a third oblique to both the former. 

6. Construct a spherical triangle in which two sides are equal and each 

of the sides oblique to the other two. 

7. Pass a great circle through three given points. 

8. Given a spherical triangle ; describe its polar. 

9. Given a spherical triangle ; draw a perpendicular from one angle 

to the side opposite. 

10. Describe a spherical triangle each of whose sides and angles exceeds 

90°. 

11. Given a = 43° 25', c = 68° 13', R = 135° 47', to find its polar. 

12. Given A = 115° 15', B = 125° 25', C= 137° 37', to find its polar. 

13. Find the polar of the triangle A = 1° 5', a = 3° 7', b = 175° 2'. 

14. Find the polar of the triangle A = 188°, a = 187°, b = 5°. 



CHAPTEE IV. 



OF RIGHT ANGLED SPHERICAL TRIANGLES. 




62. Let ABC be a spherical triangle in which C is a 
right angle. The side c opposite the right angle is 
called the hypothenuse, the other two sides a and b, 
the legs of the triangle, and 
A and B, the oblique angles. 
Let be the centre of the 
sphere, then will OA, OB, OC, 
be radii, whose lengths are 
each equal to unity. From 
B drop the perpendicular BD 
to the radius 00, and from 
D a perpendicular to the 
radius OA, and join B and E; then will BE also be per- 
pendicular to OA (Geometry). Since EB and ED arc 
respectively in the planes A OB and AOC, and perpen- 
dicular to their line of intersection, the angle between 
them is the angle of the dihedral, and therefore equal to 
A, an angle of the triangle. According to article 30, we 
have 

BD = sin a, OD = cos a, BE = sin c, 0E= cos c ; 

and according to article 5, and the values just written, 
ED = EB cos A = sin c cos A (a) 

ED = BD cot A = sin a cot A (b) 

ED = GE tan b = cos c tan & (c) 

£Z) = OD sin & = cos a sin b (d) 



86 TRIGONOMETRY. 

Placing equations (a) and (b) equal, and reducing by 
means of equation (2), we have 

sin a = sin c sin A (123) 

similarly, sin b = sin c sin B (124) 

Equation (123) is written from (124) by substituting b 
for a and B for A. This may be done, for in the former 
equation a is one leg of the triangle opposite the oblique 
angle A ; but as it is immaterial which leg and angle 
opposite are used, we may substitute one set for the 
other. 

From equations (a) and (c), 

cos A — tan b cot c (125) 

similarly, cos B — tan a cot c (126) 

From (a) and (d), 

sin c cos A — cos a sin b (e) 

From (6) and (d), 

sin & = tan a cot ^4 (127) 

similarly, sin a = tan 6 cot B (128) 

From (c) and (d), 

cos c = cos a cos & (129) 

Substituting (127) in (e) gives 

cos A — cos a sin B (130) 

similarly, cos B = cos & sin ^ (131) 

Multiplying these together and comparing with (129) 

gives 

cos c = cot J. cot .5. (132) 



63. These formulas may be associated with the corresponding ones of 
plane triangles, and thus be more easily remembered. If the arcs be 
very short, we have approximately, sin a = a, etc., tan a — a, etc., 
cos a = 1, nearly, etc. 



BIGHT ANGLED SPHEBICAL TBIANGLES. 



87 



In plane right triangles. 



sm A — — 
c 



cosA = — 
c 



tan A — — 
sin A — cos i? 



sin B — — 



COS B : 



a 

c 



tan B = — 
sin i? = cos J. 



C 2 = ^ + £2 

1 = cot A cot i? 



In spherical right triangles. 



sin A — 
cos J. = 
tan A = 
sin J. = 



sm a 
sin c 
tan & 
tan c 
tan a 



sin i? = 
cos B = 
tan j? — 
sin B — 



sin 6 

cos B 

cos & 
cos c = cos a cos & 
cos c = cot A cot 5 



sin b 
sin c 
tan a 
tan c 
tan 5 
sin a 
cos J. 
cos (2 



64. Napier's circular parts. The above equations 
are all that are necessary for the complete solution of 
right triangles. Baron Napier, an eminent mathema- 
tician, made them so symmetrical 
that they may all be stated by two 
simple rules. Taking the comple- 
ments of the two oblique angles, 
90° -A = A', and 90° - B = B', and 
of the hypothenuse, 90° — c = c, the 
five quantities a, b, c', A', B, are 
called Napier s circular parts. It 
will be found by examining any right triangle, 
neglecting the right angle that if any 
three parts be taken at random, one 
will either be between two adjacent 
parts or between two parts as remote 
as possible from it. The two former 
are called the adjacent extremes, the 
two latter the opposite extremes, and 
the third the middle part. The parts may be arranged 
within a circle as in the annexed figure, by means of 




ifter 




88 TBIG ONOMETR T. 

which the middle part and the adjacent extremes, or 
opposite extremes, as the case may be, may be readily 

selected. 



EXERCISES. 

1. Name Napier's circular parts. 

2. If a, b y comp. A, be given, which will be the middle part? and will 

the extremes be adjacent or opposite? 

3. If c be the middle part, which will be the adjacent extremes, and 

which the opposite extremes? 

4. If b, B, c, be given, which will be the middle part, and which the 

extremes? Will the extremes be adjacent or opposite? 

5. If a, B, A, are the parts, which is the middle part? and which the 

extremes? Are they adjacent or opposite? 



Napier's Eules. 

65. Substituting for A, B, c, respectively 90° — A\ 
90° — B\ 90° — c', and reducing, the preceding equations 
may be written and arranged as follows : 

sin a — cos c cos A' = tan b tan B 
sin b — cos d cos B r = tan a tan A' 
sin c — cos a cos b = tan A' tan B' 
sin A'= cos a cos B = tan b tan c 
sin B— cos b cos A' = tan a tan d 



(C) 



These stated as follows are Napier s Bales: 

TJie sine of the middle part equals the product of the cosines of the 
opposite extremes. 

The sine of the middle part equals the product of the tangents of the 
adjacent extremes. 

To aid the memory in recalling these rules, it may be 
observed that the vowels are the same in the words sine 
and mid, and cos and op, also in the words ta.?i and a.dj. 



BIGHT ANGLED SPHERICAL TRIANGLES. 89 

EXERCISES. 

1. If a = 120°, b — 100°, and B is to be found, which equation of group 

(C) should be used, and what are the circular parts? 

2. If A = 170°, B = 175°, and c is to be found, which are the circular 

parts and which formula should be used ? 

3. If c = 90°, find A and B, also a and 6. 

4. If A = 45°, and B = 45°, find c. 

5. If b = 45°, and A = 45°, find B. 

6. If B =z 135°, and b ~ 45°, find a. 

7. If ^ = 90% and a = 20°, find A and &. 

8. If the five parts are — the oblique angles, the hypothenuse, and 

the complements of the sides; deduce two rules similar to 
Napier's. 

[They are called Maudit's Rules.] 



66. The species of the parts are their relations to 90°. 
If two parts are both less than 90° they are of the same 
species; also if both are between 90° and 180 D . But if 
one is between J and 90°, and the other between 90 D 
and 180°, they are of different species. "When the result 
is found in terms of the sine of the angle, it will be am- 
biguous unless the particular angle can be determined 
by its relation to the given parts. 

In a right angled spherical triangle, if the hypothenuse is 
less than 90°, the tico sides and the two oblique angles are, 
respectively, of the same species. 

For, from the third of group (C) we will have c < 90°, 
and sin c positive ; hence 

cos a cos b 

must be positive, and therefore cos a and cos b must 
have the same sign ; hence if a is less than 90° b must 
also be less. Similarly, from the same equation, 

tan A' tan B r , or cot A cot B, 



90 TBIGONOMETRY. 

must be positive and A and B must both be less or 
both greater than 90°. 

In a right spherical triangle, if the hypothenuse exceeds 
90° the two sides and the two oblique angles will, respectively, 
be of different species. 

For, in the third of group (C), sin c' = cos c will be 
negative, hence cos a and cos b will have opposite signs ; 
and similarly in regard to A and B. 

In a right spherical triangle, an angle and its opposite 
side are of the same species. 

For, by the fifth of group (C), we have 

. cos B N 

sin A = =-, (133/ 

COS 

and since sin A will be positive (A < 180°), cos B and 
cos b must have the same sign, and hence the angles 
will be both less or both greater than 90°. 



The solution of right spherical triangles may be 
classed under six cases, as follows : 

67. Case I. Given the hypothenuse and a side, as c and a. 
To find A. Here a is the middle part, and from the 
first of group (C) we have 

, sin a 

sin A = -i 

sin c 

or, log sin A = log sin a — log sin c (134) 

To find B. In this case B is the middle par 1 and the 
fifth of group (C) gives 

cos B = tan a cot c 
or, log cos B = log tan a + log cot c (135) 



BIGHT ANGLED SPHERICAL TRIANGLES. 91 

To find b. Here c is the middle part, and the third of 
(C) gives 

, cos c 

cos b — 

cos a 

or, log cos b = log cos c — log cos a (136; 

Check. The third of (C) gives 
cos B = cos b sin A 
or, log cos B = log cos b + log sin ^4 (137) 



EXAMPLES. 

1. Given a = 20° c = 140°. 

By (134) By (135) By (136) 

log log log 

a- 20 3 sin 9.531052 tan 9.561066 ar co cos 0.027014 

c = 140° ar co sin 0.191932 cot 10.076187/1 cos 9.88425491 



A, sin 9.725984 B, cos 9.637253ft 5, cos 9.911268/1 

Gftee*, By (137), .4, sin 9.725984 

B, cos 9.637252/1 

which being the same as before found checks the 
work. Taking the angles from a table of log- 
arthmic functions gives 
Ans. A = 32^ 8 48", B = 115° 42' 23", b = 144° 36' 28''. 

[n implies that the function is negative; thus cot 140° is 
negative. Since the sign of the result depends upon the num- 
ber of negative factors, if there be but one n the result will be 
negative, and, similarly, if there be an odd number of n's in 
the logarithms added, the result will be negative ; but if there 
be an even number of n's the sign of the result will be positive. 

The last figure or figures of the logarithm of the "check" 
may not agree exactly with that previously found, and yet the 
work may be correct, for the last figure of a logarithm in the 
table is not always correct.] 



92 TRIGONOMETRY. 

2. Given a = 141° 11', c = 127° 12'. 

Ans. A = 128° 5 54", B = 52° 21' 45", & = 39° 6' 23". 

3. Given b = 18° 01' 50", c = 86° 51'. 

^/is. A = 88° 58' 25 ', jB = 18° 03' 32", a = 86 D 41' 14". 



68. Case II. Given one angle and its opposite side, as 
A and a. 

To find c. By Napier's Bules, 

sin a 





sin c — . , 


(138) 


To find b, 


sin b = tan a cot A 


(139) 


To find B, 








. ^ cos J[ 

sm B = 


(140) 



Cheeky 

sin b — sin c sin B (141) 

In this case the ambiguity is not removed by the 
tests in article 66, and hence there will, generally, be 
two solutions. 



EXAMPLES. 

1. Given B = 150°, b = 160°. 

Ans. c = 136° 50' 23" ) ( c = 43° 9' 36" 
a= 39° 4' 50" tor J a = 140° 55' 9" 
A = 67° 9' 42" j [A = 112° 50' 17" 

According to article 66, a and ^4 must be in the same 
quadrant. 



RIGHT ANGLED SPHERICAL TRIANGLES. 93 

2. Given A = 37° 28', a = 35° 44'. 

Ans. c = 73° 45' 16" ) ( c = 106° 14' 45" 

5 = 77° 54' tor J 5 = 102° 6' 

6 = 69° 50' 24" ) ( 6 = 110° 9' 36" 

3. Given A = 104° 59' a = 129° 33'. 

Ans. c = 127° 2' 27" ) ( c = 52° 57' 33" 
B = 23° 57' 19" lor J B = 156° 2' 41" 

6 = 18° 54' 38" ) { b = 161° 5' 22" 

4. Given B = 80° 1', & = 67° 36'. 

5. Given 5 = 45°, b = 45°. 



69. Case III. Ctiven the hypothenuse and one angle. 
[The student may select the formulas for this and the 
remaining cases from group (6').] 

EXAMPLES. 

1. Given A = 23° 28', c = 145°. 

Ans. B = 109° 34 33", a = 13° 12' 12", b = 147° 17' 15". 

2. Given 5 = 50° 8' 21", c = 32° 34. 

Ans. A = 44° 44', a = 22° 15' 43", 6 = 24° 24' 19". 

3. Given B = 80° 55' 27", c = 98° 6' 43". 

^ns. J = 131° 27' 18", a = 132° 6', & = 77° 51'. 



70. Case IV. Given an angle and a side adjacent. 



EXAMPLES. 



1. Given a = 118° 54', B = 12° 19'. 

Ans. c = 118° 20' 20", A = 95° 55' 2", b = 10° 49' 17" 



94 TBIGONOMETBY. 

2. Given b = 54° 30', A = 35° 30'. 

Ans. c = 59° 51' 20".8, a = 30° 8' 39".2, B = 70° 17' 35". 

3. Given B = 137° 24' 21", a = 29° 46' 08". 

Ans. b = 155° 27' 54", c = 142° 9' 13", A = 54° 1' 16". 



71. Case V. Given the two sides, a and b. 

EXAMPLES. 

1. Given a = 56° 34', b = 27° 18'. 

^ns. ^ = 16° 50' 47", B = 31° 44' 9", c = 60° 41' 9". 

2. Given a = 144° 27' 03", b = 32° 8' 56". 

Ans. A = 126° 40' 24", 5 = 47° 13' 43", c = 133° 32' 26" 

3. Given a = 32° 9' 17", b = 32° 41'. 

Ans. A - 49° 20' 17", B = 50° 19' 16", c = 44° 33' 17". 



72. Case VT. Given the two oblique angles, A and B. 

EXAMPLES. 

1. Given A = 91° 11', B = 111° 11'. 

Ans. c = 89° 32' 29", a = 91° 16' 8", b = 111° 11' 16" 

2. Given ^ = 67° 54 47", B = 99° 57' 35". 

Ans. a = 67° 33' 27", b = 100° 45', c = 94° 5'. 

3. Given A = 54° 01' 15", B = 137° 24' 21". 

Ans. a = 29° 46' 08", b = 155° 27' 55", c = 142° 09' 12". 



73. A quadrantal triangle is one having one or more 
of its sides equal to a quadrant. Its polar triangle will 
be right angled ; hence to solve a quadrantal triangle 
pass to its polar, solve that, and pass back to the origi- 
nal triangle. 



RIGHT ANGLED SPHERICAL TRIANGLES. 95 
EXAMPLES. 

1. Given c = 90°, B = 74° 4,5', a = 18° 12', to find the 

remaining parts. 
Am. G = 104 J 31' 13", b = 4° 42 15 ", A = 71° 10' 54". 

2. Given c = 90°, A = 42° 01', B = 121° 20' to find the 

remaining parts. 
Ans. C = 66° 57' 15", b = 111° 50' 18", a = 45° 40' 17". 



74. If the angle sought be near zero, or ninety degrees, it may some- 
times be found more accurately with ordinary logarithmic tables by 
means of special formulae, a few of which are here given without proof. 
tan 2 (45° — \A) — tan } (c — a) cot $ (c + a) 
tan 2 %b = tan J (c — a) tan \ (c + a) 

tan 2 \B = sin (e - a) esc i (c + a) [56, 131, 30 ~ 31 J 

tan 2 ic = - cos (A + 5) sec (A - B) [56, 129, 32 -f- 33] 
tan 2 (45° - he) = tan £ (J. — «r) cot i ( J. + a) 

To obtain the first of these, we have from equation (123) 

. sin a 
sin J. = - — ; 
sm c 

... l-sin^l = sin e- sin a = ^ fl _ fl) eot } (c + B> 
1 + sm ^4. sin c + sm a 

In equation (66) make x - 90°, and it will readily be found that the 
left member of the above equation reduces to tan 2 (45° — \A). 

The fifth equation above maybe found in a similar manner by begin- 
ning with equation (123) after writing it thus : 



OXJ.J. 1/ -J j • 

sm A 

To obtain the second equation above : 

ri7 /eat i 4-2 17, 1 — 2 cos 6 + cos 2 b 
[Eq. (56) ] tan 2 \b = 1 _ CQg8 h , 

[Eq. (129) ] cos 2 \b = cos 2 c -s- cos 2 a, 

which substituted in the preceding and reduced by the aid of (67) mul- 
tiplied by (69) will at once give the required result. 



CHAPTEE V. 



OBLIQUE ANGLED TRIANGLES SOLVED BY MEANS OF RIGHT 
ANGLED TRIANGLES. 

75. In a spherical triangle the sines of the angles are pro- 
portional to the sines of the sides opposite. 

In the spherical triangle ABC, let fall the perpen- 
dicular BD from any angle B upon 
the opposite side, AG, thus forming 
two right angled triangles ABB and 
CDB ; of which let m be the base of 
one, and n that of the other, x the 
angle opposite m and y that opposite 
n, as in the figure. According to 
Napier's rules, we have for the 




triangle ABB, 
triangle CBB, 



sin p — sin c sin A 
sin p — sin a sin G 



Therefore, 

sin a sin G — sin c sin A 

similarly, sin b sin A = sin a sin B 

sin c sin B — sin b sin G 



(142) 
(143) 



(144) 



These may be written 



sin a __ sin b _ sin c 
sin A ~~ sin B ~ sin G 



which was to be proved. 



(145) 



OBLIQUE ANGLED TRIANGLES. 97 

76. To find the segments m and n, given the three sides. 
From Napier's first rule, 

cos a = cos p cos n 

cos c == cos p cos m 

Divide the former by the latter ; then after adding 
each term to unity, and subtracting each term from 
unity, divide the latter by the former, and find 

cos c — cos a _ cos m — cos n 
cos c + cos a ~~ cos m -f cos n 

But, (69), 

cos c — cos a 
cos c + cos a 
and 

cos m — cos n 



— tan | (a + c) tan i (a — c) 
tan -J (ft + m) tan % (n — m) 



cos ?w -t cos n 
.\ tan | (a + c) tan ^ (a — c) = tan 46 tan \ (n — m) (146) 

from which n — m may be found ; and since n + m = b, 
the values of n and m become known. 

By means of Napier's rules and equation (146) all the 
cases of oblique triangles may be solved, and the solu- 
tion may sometimes be facilitated by means of equa- 
tions (144). 

The perpendicular BD may fall entirely without the triangle, and if 
it falls to the right of BC, n will be entirely on the prolongation of the 
base, and m will be the entire base phis the elongation. This perpen- 
dicular will also meet the great circle of the base in two points, distant 
from each other 180°, and if that perpendicular be taken whose foot is 
nearest to one end or the other of the base, then will m + n be numeri- 
cally less than 180°, as it should. If n be the external segment, it will 
be considered as minus, so that, algebraically, we will have in all cases 
m + n — b, which, numerically, may sometimes be m — n = b. 

Since the greater segment is not necessarily adjacent to the greater 
7 



98 



TRIGONOMETR Y. 



side, its position must be determined. This may be done by means of 
equation (146) ; for the signs of all the factors except the last will be 
determined from the data, and therefore the sign of tan i (n — m) be- 
comes known, and since i (n — m) will be numerically less than 90°, it 
follows that when tan | (?i — m) is + , n > m t and if it be — , n < m. 



In the solution of oblique spherical triangles, we may 
have six cases, as follows : 

77. Case I. Given two sides and the included angle, as 
a, b, C. 

From the angle B drop the perpendicular p on the 
side opposite ; then find p by means of equation (143). 

Find the segment n by means of the second of Na- 
pier's rules (observing the species), which will give 

cos C (M7) 




tan n = 



m 



tan a* 

b — n 



(148) 



Knowing m and p, find A by Na- 
pier's first rule, and c by the second, 
giving 

sin on 



cot A = 



cos c 



tan p" 

cos p cos m 



(149) 
(150) 



Find B by means of (144) 2 ; or find 
x and y by means of the right triangles, in which case 

B = x + y. 



78. Case II. Given tivo angles and the included side. 
Pass to the polar and solve by Case I., then pass 



OBLIQUE ANGLED TRIANGLES. 99 

back. Or, given C y B, a, by means of Napier's rules, 
find n, p, y. ; then x = B — y, and with x and p complete 
the solution. 



79. Case III. Given two sides and an angle opposite 
one of their/, as a, c, A. 

Find the angle C by means of (144)i. Then by Na- 
pier's rules 

cos A = tan c tan m (151) 

cos C = tan a cot n (152) 

by means of which m and n may be found, then 

b = m + n, 

after which B may be found by means of equation (144) 2 . 

This case admits of two solutions, one or none, which 
may be shown in the same manner as in article 52, Case 
II., of plane triangles. 

If a > p, and a < c, there will be two triangles. 

If a = p, there will be one triangle. 

If a < p, there will be no triangle. 

If a = c, there will be one triangle. 

If a > c, and ^4 < 99° or A > 90°, there will be one 
triangle. 

If a > c and A = 90°, there will be two triangles. 



80. Case IV. Given two angles and a side opposite one 
of them, as A, C, a. 

Pass to the polar, solve that by Case III., and then 
pass back. Or, solve directly, finding c by (144) 1? m and 
n by (151) and (152) ; then b — m + n, after which B 
may be found by (144)* 



100 TRIGONOMETRY. 

81. Case V. Given the three sides. 
By means of (148) we have 

tan -h (n — m) — tan i (a + c) tan -A- (a — c) cot i (ft + ra), 

(153) 

from which find n and m ; after which the solution may 
be computed by Napier's rules. 



82. Case VI. Given the three angles. 
Pass to the polar, solve that by Case V., and then pass 
back. 



EXAMPLES. 

1. Given A = 92° 10', B = 72 D 15', and C = 135° 15' 
20", to solve the triangle, 
This comes under case VI. , and we have, passing to 
the polar, 

a' = 87° 50', V - 107° 45', c =-- 44° 47' 40". 

By (153) 
\ (a + c') = 66° IT 20" log tan 10.357337 

^(a'-c')r= 21° 32' 40" log tan 9.5K'6384 

i (n + m) = 53° 52' 80" log cot 9.863252 

■K?i-m)= 33° 16' 9" log tan 9.816973 

By (125) By (125) 

.'. n = 87° 08' 39" log tan 11.302022 

m = 20° 36' 21" log tan 9.575178 

a' = 87° 50' 00" log cot 8.577877 

c' = 44° 44' 40" log cot 10.003875 

C = 40° 40' 35" log cos 9.879899 

A' = 67° 42' 21" log cos 9.579053 



OBLIQUE ANGLED TBI ANGLES. 101 

To find B'. 

By (132) By (132) 

a' = 87° 50' 00" log sec 11.422434 

C = 40° 40' 35" log cot 10.065795 

c = 44° 44' 40" log sec 10.148587 

A = 67° 42' 2" logcot 9.612795 



y = 88° 08' 21" log tan 11.488229 

x = 29° 59' 48" log tan~9.761382 

C =118° 08' 09" 

Passing back we find 

(a =112° 17' 39" 

Ans. J b = 139° 19' 25" 

( c = 61° 51' 51" 

2. Given a = 62°, b = 75°, tf= 100\ 

3. Given A = 150°, 5 = 60° C = 120°. 

4. Given ^ = 50° 12', 5 = 58 08', a = 62° 42', to find 

the other parts. 

( b = 79° 12' 10", c = 119= 03' 26", G = 130° 54' 28" 
100° 47' 50", c = 152° 14' 18", G = 156 3 15' 6" 



(orb = 



5. Given a = 90°, b = 90°, c = 90°. 

6. Given a = 45°, b = 45°, c = 45°. 

7. Given a = 175°, ft = 175°, c = 5°. 

8. Given a = 84° 14' 29'', 6 = 44° 13' 45", and ^4 = 130° 

5' 22" to find the other parts. 
Ans. B= 32° 26' 6§", (7= 36° 45' 28", c = 51° 6' 12". 

9. Given the two sides 44° 13' 45" and 84° 14' 29", and 

the included angle 36° 45' 28" ; to find the other 
parts. 
Ana. Angles 32 D 26' 6", and 130° 5' 22", side 51° 6' 12". 



CHAPTEE VI. 

GENERAL FORMULA. 

83. Value of the cosine of a side of a spherical triangle. 
Let ABC be a spherical triangle, the centre of the 

sphere, — ABC the tri- 
hedral. Conceive the tri- 
hedral to be cut by a plane 
perpendicular to the edge 
OA, passing through any 
point D, and let DF, DE, 
EF, be the intersections 
with its faces. Then will DF and DE be perpendicular 
to OA, the triangles ODF and ODE will be right, and 
the angle EDF = A. 

"We have, article 55, 

EF 2 = OE 2 + OF' -WE . Oi^cos a (a) 

EF 2 = DE 2 + DF 2 - 2DE . DF cos A (6) 

OD 2 = 0^ 2 - DJE" (c) 

OD 2 = OF 2 - DF 2 (d) 




Subtracting (6) from (a) and reducing by (c) and id), 
gives 

Oi) 0/) Z># DF 



cos a 



0# OF V OE' OF COS J ' 



GEXERAL FORMULA. 103 

Substituting the trigonometrical functions, group (A), 

cos a — cos b cos c -f sin b sin c cos A \ 

cos b = cos c cos a * sin c sin a cos i? V (154) 

cos c = cos a cos 6 + sin a sin b cos (7 J 

To aid in memorizing these, observe that the first and last are cos of 
letter of same name, thus cos a and cos A, the former small the latter 
large ; also that cos A is joined to the factors of sines of the other sides. 

Also observe that the second equation may be written from the first 
by advancing the letters one in the scale a, b, c, a; thus, for a write 
b, for b, c, and for c write a. Similarly the third may be found from 
the second. 



84. Let ABC be a spherical triangle polar to 
then by geometry, 

A'=180°-a a =180° -A 
5=180° -6 6=180 -B 
C'=180 z -c c'=180°-G 



ABC, 




The firs': of (154) applied 
to the triangle ABC\ 

gives 

cos a = cos b' cos c' -f sin b' sin c cos A\ 

in which substituting the above values, we have 

— cos A = (- cos B) ( — cos C ) 4- sin B sin C ( — 

or 

cos A — — cos B cos C + sin B sin C cos a 
cos B — — cos C cos J. + sin C sin 4 cos 6 
cos C — — cos J. cos B -f- sin 4 sin B cos c 



cos a) 



(155) 



104 TRIGONOMETRY. 

Thus, by the use of the polar triangle, formulae may 
be obtained in which the functions of the angles and 
sides may be found in place of the sides and angles. 



85. Substituting cos a from (155),, in (154) 2 , we have 

cos b — cos 2 c cos b + sin b cos c sin c cos A 
4- sin a sin c cos B 

But, 

cos 2 c cos b — (1 — sin 2 c) cos b = cos b — cos b sin 2 c 

which substituted and dividing by sin c, gives 

sin a cos B — sin c cos 6 — sin b cos c cos J. 

sin b cos (7 = sin a cos c — sin c cos a cos B ^ (156) 

sin c cos J. = sin 6 cos a — sin a cos b cos (7 



86. From (144) 2 we have 

sin a sin S 



= sin b 



sin J. 
which, divided into (156)! member by member, we have 

sin A cot B = sin c cot & — cos c cos A \ 
and, sin J? cot C = sin a cot c — cos a cos 5 >- (157) 
sin C cot J. = sin b cot a — cos b cos (7 J 

In the triangle ABC, A and B may be interchanged 
provided that a and & are also interchanged. Inter - s 
changing these letters in the precedi g equations, gives, 



GENEBAL FOEMULM. 105 

sin B cot A — sin c cot a — cos c cos B \ 

sin' JL cot = sin & cot c — cos 6 cos A V (158) 

sin C cot i? = sin a co': & — cos a cos (7. J 

. In these equations the permutations may be made by passing back- 
ward with the letters, thus, (7, B, A, C, or by beginning with the third 
equation and writing the 2d from it by permuting the letters in their 
natural order, and the first from the second in the same manner. ■ 



87. To transform (154) and (155) so as to adapt them 
to logarithmic computation. 
In (154) substitute 

Eq. (46), cos A . = 1 - 2 sin 2 \A 

then 

cos a = cos (6 — c) — 2 sin b cos c sin 2 i A 

. , . A cos (6 - c) - cos a , Q , 

.'. 2 sin \A — . 7 . (loy) 

sin hmc 

If x = a, and y = b — c 

then 

£ (# + y) = i (a -f 6 — c), i (» - ^) = i (a - 6 + c) 

and(Eq. (65)), 

cos (6 — c) — cos a = 2 sin 2 i (a — 5 + c) sin 1 (a — 6 + c), 

which substituted in (159) gives 

. , , A sin | ( a— 6 + c) sin | (a + 6 — c) « flA . 

sin M = = — $—. (160) 

sin b sin c v ' 

Let s be one half the sum of the sides, that is, 
• a + b -f c - 2s 



106 



TRIGONOMETRY. 



then a— b + c = a + b — 2b + c = 2(s — b) 
a + b — c = a + b + c — 2c — 2 (s — c) 
which substituted in (160) gives 

sin (s — b) sin (s — c) 



sin 2 \A = 



similarly, sin 2 \B = 
sin 2 i(7 = 



sin b sin c 




sin (s — c) sin (5 


— a) 


sin c sin a 




sin (s — a) sin (5 


-b) 



sin a sin b 



Passing to the polar triangle, 

sin{S- B) sin (S ■ 



cos 2 ia 



cos 2 -J6 = 



cos 2 ic 



G) 



sin J? sin G 

sin (5 - C) sin ff - ^1) 
sin (7 sin J. 

BinQS f -^ )sin( /S r -.g) 
sin A sin 5 



(161) 



(162) 



Again, substituting, 
Eq. (47), cos A = 2 cos 2 ±A - 1 

in (154)i gives 

cos a — cos (6 + c) + 2 sin 6 sin c cos 2 -M. ; 

cos a - cos (6 + c) 



cos 2 iM. = 



2 sin & sin <^ 



or 



cos 2 IA = 



_ sin 5 sin (5 — a) 
sin 6 sin c 



similarly, cos 2 \B = 



_ sin 5 sin (5 — 6) 



sin c sin a 



cos 



1 r* _ s ^ n s s i n *( 5 — c ) 

sin a sin 6 



(163) 



GENERAL FORMULA. 

Passing to the polar triangle, 

— cos 8 cos (8 — A) " 



sin 2 \a — 



sin B sin C 



. , , _ — cos # cos (8 — B) 

sin -Jo = ri i 

1 cos 6 cos :A 

. 2 , — cos 8 cos ( # — (7) 

sm f c = i ^ 

cos- J. cos B 



107 



(164) 



88. From (161) and (163) we find, observing that 

sin %A -r- cos \A — tan ^1, 

tan' U = ™(«-») ■»(«-') 

sm 5 sm (s — a) 



tan 2 W = ■*»(«-«) ■*(«-«) 

sin 5 sm (s — ft) 

tan-|0^ sin(s - a)s / in( ' 9 ~ ? ' ) 
sm 5 sm (s — c) 



(165) 



Passing to the polar triangle ; or by means of (164) 
and (162) we have 

- cos 8 cos (8 — A) 



tan 3 ia = 



tan 2 -J6 = 
tan 2 Ac =-- 



cos (8 - B) cos (8 - C) 

— cos 8 cos (# — B) 
~ wsJS - (7) cos (8 - A) 

— cos 8 cos (# — C) 



cos (# - ^) cos (8- B) J 



(166) 



89. Where several parts — as for instance all the 
angles — are to be found, it may be better to proceed as 
follows : 



108 

Let 



TBIQONOMETR T. 



t _ ,j/ sin (s - o) sin ( s - b) sin (s - c) 

sin s \ J 

then (165) become 

tan \A = 

tan 15 = - (168) 



sin (5 - 


- a) 


Tc 




sin (s - 


-i) 


% 





tan^C = -.- 

sin (s — c) J 



Similarly, put 



^ 



— COS # 



(169) 



cos (S - A) cos (S - 5) cos (S-C) 
then (166) become 

tan \a = K cos (# — A) 
tan |6 = Keos(S-B) } 
tan fc = J5Tcos(£- (7) 



90. To deduce Napier's Analogies, which are 
tan ic sin % (A + B) * 



tan i (a — b) 
cot JO 

t2LKi(A-B) 

tan |c 

tan i (a + 6) 

cot*C 



" sin + (J. - if) 
sin i (a 4- 6) 
sin i (a — b) 
cos !(^L + B) 

: cos i (A - 5) 
cos J- (a + &) 



tan i (J. + B) cos i (a — b) m 



(171) 



GENERAL FORMULAE. 109 

Dividing (165)! by (165) 2 gives 

tan \A _ sin (s — b) 
tan \B ~ sin (s — a) 

By composition and division, we have 

tan -J A 4- tan |i? sin (s — b) + sin (5 — a) 
tan £ J. — tan %B ~ 

Equation (66), = 

~~ tan I {a — 6) 

To reduce the left member, we have from (30) and (31) 
sin (x 4- y) sin x cos ?/ 4- cos x sin ?/ 
sin (x — y)~ sin a; cos y — cos x sin y' 
and dividing both numerator and denominator of the right member by 
cos x cos y we find 

sin (x 4- y) _ tan x 4- tan y 
sin (x — y)~ tan a: — tan 2/ 

Making aj —\A, and y — \B, we have 

tan I J. + tan £2? _ sin \ (A + B) tan £c 



sin (s — b) — 


sin (s 


-a) 


tan £ (« — b 


+ s- 


a) 


tan £ (s — • b - 


-(•- 


en) 


tan \c 







tan iJ. - tan iB sin i (A - B) *~ tan | (a - J) 
which is (171)i. 

Passing to the polar triangle we have 

sin j(a 4- b) cot \C 



sin \(a — b) tan J ( A — i?) 
which is (171) 2 . 

Multiplying (165), by (165) 2 , 

tanUtan^? = sln f S - c) 
sin 5 

or, 1 : tan £4 tan ^Z? : : sin s : sin (s — c\ 

which by composition and division, 

1 — tan \A tan \B _ sin s — sin (s — c) 
1 4- tan \A tan ^Z? ~* sin s 4- sin (s — c) 



110 



TRIGONOMETRY. 



Equation (66), 

hence 

which is (171) 3 . 



tan ic 



tan \ (a + b) 

cos i ( A + B) _ tan \c 
cos i (A — B) ~ tan \ (a + 6) 



Passing to the polar gives 

cos | (a + &) 



cot £ 



cos | (a — b) tan £ ( A + 5) 



which is (171) 4 . 



These equations are especially useful in solving a triangle in which 
two sides and the included angle, or two angles and included side, are 
given. In using them the species of the parts must be observed. In 
the first, tan i (a — b) and sin | (A — B) are necessarily of the same 
species, and therefore tan \c and sin \{A + B) must be of the same 
species. Similarly, in the second, tan \ ( A — B) and sin \ (a — b) are of 
the same species, and therefore cot \G and sin \ (a + b) must be of 
the same species. 



By means of the general equations the solutions of 
the six cases may be made as follows : 

91. Case I. Given two sides and the included angle, as 
B a, 6, C, to find the remaining 
parts. 

To find the angles A and B. 
From (171) 2 and (171) 4 we 
C have 




tan $(A-B) = 



tan J(^.+ £).= 



sin 


i(a- 


h) 


sin 


*(« + 


V) 


cos 


h {a — 


-b) 



cos £ (a + b) 



cot 10 



cotiC 



(173) 

(174) 



GENERAL FORMULA. HI 

by means of which i (A — B) and i {A + B) may be 
found, and of the last two expressions the sum is A, 
and the difference B. 

To find c. From (144)i we have 

sin 6 . ., „ x 

sin c — -. -j sm a (175) 

sm A x J 



EXAMPLE. 

Given a = 73° 58', b = 38° 45', (7 = 46° 33' 39". 

Then to find A and B 

By (174) - By (17S) 

! ( a _ h) = 17° 36' 30" log cos 9.979160 log sin 9.480738 

%(a + b)= 56° 21' 30" log sec 0.256493 log esc 0.079606 

| G = 23° 16' 49".5 log cot 0.3662C6 log cot 0.366266 

i (A + B) = 75° 57' 33". 8 log tan 10.601919 

1{A-B)= 40° 10' 54".3 log tan 9.926610 

^ = 116° 8'28".l 
B - 35° 46' 39". 5 

To find c 

By (175) 

G= 46° 33' 39" log sin 9.861000 

a= 73° 58' log sin 9 982769 

A = 116° 8' 28". 1 log esc 0.046863 

c = 51° 00' 15". 8 log sin 9.890532 

^4 = 116° 8'28".l 

\B = 35° 46' 39".5 

c- 51° 00' 15".8 



92. Case II. Given two angles and tJie included side, 
A, B, c. 

Pass to the polar, solve that by Case L, and then pass 
back. 



112 TRIGONOMETRY. 

Or, by means of (171)i and (171) 8 find a and b, thus : 

tan i {a - b) = ^ | ^ + ^ tan £c (176) 

tan * ( a + 6 ) = cosi(J + S tan * C (177 ^ 
Then find <7by(144) 2 , 

sin C=*^smA. (178) 

sin a v ' 



93. Case III. Given two sides and the angle opposite 
one of them, as a, b> A. 

Find B from (144) 2 , then c from (171)t and G from 
(171) 2 . The equations are 

sin B — sin A sin 6 esc a (179) 

tan * C = sini(^i ) tan * (a " 6) (180) 

cot i(7 = S ! n f i a + -§ tan \ {A - B) (181) 
sin |- (a — 6) y 

Equations (154) will give c and C directly by means 
of natural functions. 

This case, like the corresponding one in plane trian- 
gles, may have two solutions, one solution, or no solu- 
tion, as already shown in Article 79. There will be 
two solutions when 

A<90°, a + 5<180°, a<b, 

or A > 90°, a + b > 180°, a > b. 

If sin B — 1 in (179), there will be one solution, and 
the side a will be perpendicular to the side b. 
If sin B > 1 there will be no solution. 



GENERAL FORMULA. 113 

94 Case IV. Given two angles and the side opposite 
one of them, as A, B, a. 

Pass to the polar, solve that and then pass back. Or, 
solve directly by means of (144) l5 (171)i, (171) 4 . There 
may be two solutions. 



95. Case V. Given the three sides, a, b, c. 
Solve by means of equations (165). 



96. Case VI. Given the three angles. 
Solve by means of equations (166). 



97. Area of the spherical triangle. Let A, B, C, 

be the angles of the triangle, R the radius of the sphere, 
E the spherical excess, and K the area, then 



E = A + B+ C- 180° 

E 

180° 



K = ~ttR- (182) 



EXAMPLES. 

1. Given a = 70° 4' 18", b = 63° 21 27", c = 59° 16' 23", 

to find the angles A and B. 

Ans. A = 81° 38' 20", B= 70° 9' 38". 

2. Given the three sides of a spherical triangle, 120° 43' 

37', 109° 55' 42", and 116' 38' 83"; required the 
angles. 

Ans. 115° 13' 26", 98° 21' 40", 109° 50 22". 

8 



114 



TRIGONOMETRY, 



Applications of Sphepjcal Tbigonometky. 

98. Definitions. The celestial sphere is the imaginary 
concave surface in which, all the celestial bodies appear 
to be situated. Its radius is considered as infinitely 
long. 

The sensible horizon is the circle in which a tangent 

plane to the surface of the 
earth cuts the celestial 
sphere. 

The real horizon is the 
great circle in which a plane 
through the centre of the 
earth cuts the surface of 
the celestial sphere. 

The Zenith of the observer 
is a point directly over his 
head in the surface of the celestial sphere. It is one 
pole of the horizon. Let be the place of the ob- 
server, Z his zenith, then will HBH be his horizon. 

The celestial equator or equinoctial is the great circle 
in which the plane of the earth's equator produced cuts 
the surface of the celestial sphere, as ED Q. 

The axis of the earth is the imaginary line about 
which the earth rotates. The axis of the celestial 
sphere is the same line prolonged, as, POP'. 

The poles of the equinoctial are the points where the 
earth's axis pierces the surface of the celestial sphere, 
as P and P\ 

The ecliptic is the great circle of the celestial sphere 
cut by the plane of the earth's orbit. 




GENERAL FORMULA. 115 

The equinoxes are the points of intersection of the 
ecliptic and the celestial equator. 

Hour circles or circles of declination are great circles 
passing through the poles of the equinoctial, as PSP . 

The celestial meridian of the observer is the great 
circle of intersection of the plane of the terrestrial me- 
ridian of the observer with the surface of the celestial 
sphere. 

Vertical circles are great circles passing through the 
zenith of the observer, as ZSB. 

The east and west points are where the vertical circles 
perpendicular to the meridian of the observer cut the 
horizon. 

Prime verticals are verticals passing through the east 
and west points. 

The obliquity of the ecliptic is the angle between the 
plane of the ecliptic and the plane of the equator, and 
is about 23° 27'. 

The declination of a star is its distance north or south 
of the celestial equator. If S be the place of a star JDS 
will be its declination. It corresponds to terrestrial 
latitude. 

The polar distance of a star is the complement of its 
declination, as PS. 

The right ascension of a star is the dihedral angle be- 
tween the hour circle of the star and an established 
meridian. The established astronomical meridian 
passes through that equinox which is in the constella- 
tion Aries ; and right ascension is measured from this 
meridian east through 360°. The right ascension and 
declination of celestial bodies are given in nautical 
almanacs or ephemerides. 



116 TRIGONOMETRY. 

The difference between the right ascensions of two 
stars is the angle between their hour circles. Thus, if 
S and S ' are the positions of two stars, PS and PS' 
their hour circles, then will the difference between their 
right ascensions be SPS', which is measured by the 
arc DD on the equator between the respective hour 
circles. 

The azimuth of a star is the angle between the merid- 
ian of the place and the vertical through the star, as 
HB, or HOB = the angle A at Z. 

The hour angle of a star is the angle between the hour 
circle of the star and the meridian of the place, as ZPS. 
When given in degrees, it may be reduced to hours by 
dividing by 15, or to minutes by multiplying by 4 ; for 
the earth revolves through 15 degrees in one hour, or J 
of a degree in one minute. 

The altitude of a star is its angular elevation above 
the horizon, as BS. 

The zenith distance of a star is the complement of its 
altitude, as ZS. 



99. To find the shortest distance between tivo places on the 

earth's surface given by their latitudes and longitudes. 

Let A and B be the places, DB 
the latitude of B, EA that of A, 
bB and bE, their respective longi- 
tudes ; then in the triangle APB, 
we have the angle APB — ED — 
the difference of their longitudes ; 
PB and PA the co-latitudes, or 
polar distances. The arc AB will 
be the arc of a great circle. Hence 
there will be given two sides and 
the included angle, and may be 
solved by Case L, Article 91. 




GENERAL FORMULA. 



117 



EXAMPLES. 

1. The latitude of the government post-office at New 

York City is 40° 42' 44" N., its longitude 74° 0' 
24 W. ; the latitude of Liverpool is 53° 25' N., 
and longitude 3° W., what is the shortest dis- 
tance between them in miles on the earth's sur- 
face, the earth being considered a sphere whose 
radius is 3,956 miles. 

Ans. 3,305 miles. 

2. The latitude of San Francisco being 37° 48' N., its 

longitude 122° 23 "W., and that of New York, 
that given in example 1, find the shortest dis- 
tance on the earth's surface between them. 

Ans. 2,562 miles. 



100. The latitude of a place 
equals the elevation of the pole 
above the horizon. 

For if Z be the zenith of the ob- 
server, EZ will be his latitude; and 
P being the pole of the sphere, HO 
the horizon, we have ECP= ZCO 
= 90 ? . Taking ZCP from both, 
leaves ECZ = PCO = latitude. 




101. To find the time of sunrise at any given piece on a 
given day. 

Let C be the place of the observer, HGH' his horizon, RBA the 
small circle in which the sun appears to move, P the pole, EQ the 



118 



TRIGONOMETRY. 



equator, PCH' the latitude of the observer. The sun will appear to 

rise when it comes into the horizon 
at S. Then in the triangle SLC, 
right angled at L, we have SL — 
the declination of the sun, SOL 
— the co-latitude of the observer ; 
hence, CL which measures the 
hour angle GPL, may be found 
from one of Napier's rules, giving 

sin CL = tan Led. . tan Dec. 

The sun would be six hours in 
describing the arc BA, and when 
at A it would be noon; hence CPL 
reduced to time, will be the time of rising before six o'clock. The 
same rule applies to any other celestial body. If rba be the path of the 
star, the time of rising will be six hours less the hour angle S'Pb. 




EXAMPLES. 



1. Required the time of sunrise at Hoboken on the 
longest day of the year; the latitude of the 
place being 40° 43 48". The greatest declina- 
tion of the sun will be 23° 27'. 



Led. = 40° 43' 48 ' 
Dec. = 23° 27' 

hour angle — 21° 55' 54'' 



log tan 9.935027 
log tan 9.637265 

los: sin 9.572292 



= Hi. 21m. 43.6s. which taken from 6/?. gives 
4th. 32m. 16.4s. 

2. Required the time of sunrise in the preceding exam- 
ple, on the shortest day of the year. 

Ans. Ih. 27m. 43.6-5. 

[The declination of the sun will be 23° 27' south; and the 
preceding computation will give the time after six o'clock.] 



GENEBAL FORMULAE. 



119 



3. Kequired the length of the longest day at Hoboken. 

[It will be found that the number of hours in the length of 
the day will be twice the hour of sunset ; and hence the length 
of the night will be twice that of the sunrise.] 

4. "When the declination of the sun is 23° 27', at what 

latitude will the sun not rise ? 

[If the declination be north it will rise at that place at mid- 
night, or in other words be above the horizon 24 hours ; but if 
the declination be south it will just come to the horizon at 
noon.] 

5. At what time will the sun rise for a place on the 

equator ? 

[It will be found to be independent of the declination of the 
sun, and be at six o'clock.] 

6. When the declination of the sun is 15° N., what will 

be the latitude of the place at which the sun 

rises at 4 o'clock? 

[No allowance in these examples is made for refraction, 
semi-diameter of the sun, ellipsoidal form of the earth, or 
change in declination from mean noon.] 



102. To find the time of the 
day from the altitude of the sun. 

Let C be the place of the observer, 
Sthe sun (or star), ABB its path; 
then will EZ be the latitude of the 
observer, ZP = the co-latitude, DS 
— the altitude of the sun, ZS — co- 
altitude or zenith distance, qS — the 
declination of the sun, PS — co- 
declination or polar distance, and 
ZPS = the hour angle, sought, 
which reduced to time will be the hours before noon if S be east of the 
meridian, and after noon if it be west of the meridian. In the tri- 




120 TRIGONOMETRY. 

angle ZPS, three sides are known to find the angle P. Hence equa- 
tion (163)i gives 



lvno t /sin s sm (s — a) 

cos iZPS = A/ . 7 \ 1 

V sm o sm c 

where 

2s = (90° - L) + (90 = - A) + (90° - B) = 270° -(L + A+D) 
.\ t=-185°-i(£ +-4 + -D) 

a = 90° - A, b = 90° - £, c = 90° - D. 

Example.— In latitude 40° 21' N., when the declination 
of the sun is 3° 20' S., and its altitude 36° 12 , 
what is the time of day, after noon ? 

Arts. 27*. 17m. 205. 



103. To find the azimuth at extreme elongation of a star. 

If S \ in the figure in the next article, be the star, 
then at the extreme elongation ZS P w T ill be a right 
angle, PS ' the polar distance of the star, and ZP the 
co-latitude, and by Napier's rules we have 

sin (90" - 6') = sin (90° - cp) sin A, 
or cos S' = cos cp sin A ; 

cos 6' 



sin A 



cos cp 



Example.— If 6' = 88° 41' 24", <p = 40° 43' 48", re- 
quired A. Ans. A = 2° 0' 28". 



104. Given the right ascensions of two stars and their 
declinations, to find their common azimuth angle when in the 
same vertical, and their hour angles. 




GENERAL FORMULAE. 121 

Let S and S' be the respective positions of the stars. 
cp — the latitude of the place = EZ, 
6 = the declination of 

S=DS, 
6 f = the declination of 
8' = D'S', 
a — a' ' = the difference of 

their right as- H ,[ 
censions=#P#', 
a = the side SS\ 
then 

ZP = 90°-(p = the co- 
latitude, 
P^ = 90° — S = the polar distance of S, 
PS' = 90° - tf = the polar distance of 8', 

^ = PZ£ = HOB = the azimuth. 
From equation (154)! 
cos a = sin d sin (5' + cos d cos #' cos (a — a/) (183) 

Let m and JV be auxiliary quantities and have such 
values that 

msin iV=sm (?' (184) 

m cos N = cos £' cos (ar — a-') (185) 

which may be done since by these two equations the 
two unknown quantities m and N may be found. 

Dividing (184) by (185), 

tan 6' 



tani\r = 



also 



m 



sin <?' 



sin^ 



Zt> or 



cos (or — a') ' 

cos #' c os (a- — a' 7 ) 
cos JV 



(186) 
(187) 



122 TRIGONOMETRY. 

Substituting (184) and (185) in (183) reducing and 
substituting m from (187) gives 

, _ ^ TX sin S' cos (6 — N) _ __. 
cos a = m cos (d — N) = -A^ • (188) 

From ZPS\ equation (144)^ 

. sin J. cos <p 

sm >S = ^ — ■ (189) 

cos d v J 

and from &?# ', 

. , cos S s in (a - a') 
sm S = . — ; (190) 

sin a ' v J 

dividing and reducing, gives 

cos S cos 6' sin (a — a) ,« M ^ 

sm ^ = .— - S (191) 

cos 95 sm a v 7 

which gives the required azimuth. 

To find the hour angle ZPS' = £ 
From equations (156) 

sin < cot J. = sin (90° - <p) cot (90° - (J) 
— cos (90° — <p) cos t 
— cos <^ tan d — sin <p cos t. 
Let 6 sin B == sin <?> ) 

6 cos I? = cot ^L J 



(192) 



then 6 = ^§, tan2? = 5B* (193) 

sin i?' cot JL v y 

Substituting (192) in the equation immediately pre- 
ceding, we have 

b sin t cos B -t- b sin _Z? cos £ = cos cp tan (J 
or sin (t + B) --= cot <p tan S sin i?, (194) 

by means of which t may be determined. 



GENERAL FORMULAE. 123 

EXAMPLE. 

To find the common azimuth, and the corresponding- 
hour angle for Polaris and a Ursse Majoris at the 
place whose latitude is 40° 43' 48". 5, and longi- 
tude Oh. 12m. 8s. east of the meridian of Wash- 
ington. (This is very nearly the latitude and 
longitude of Stevens Institute, Hoboken, N. J.) 

Since the right ascension and declination of the stars is con- 
tinually, though very slowly, changing, it will be necessary 
when great accuracy is desired, to ascertain these quantities 
for the date of the proposed observation. Making the compu- 
tation for January 1, 1884, we find from the Nautical Almanac 
for that date that for 

Polaris the declination was 88° 41' 24". 87, 

and the right ascension 17*. 16???. 14.295. 

and for e Ursae Majoris dec = 56° 35' 

R. A. = 12A. 53???. 

The quantities for either star may be primed. Let 8' = 88° 
41' 24". 87, 8 — 56° 35'. Then a — a' reduced to arc will be 
174° 11' 25". 6. The work may be arranged as follows: 

By (186) By (188) By (191) 

log log log 

8' = 88° 41' 24".87 tan 11.640856 sin 9.999886 cos 8.359028 

8 = 56° 35' cos 9.740334 

a - a = 174° 11' 25".6 sec 0.002236» sin 9.005275 

JT= 91° 18' 11" tan 11. 643092 w esc 0.000113 

8-1?=- 34° 48' 11" ccs 9.914844 

a = 34° 43' 11" cos 9.916843 esc 0.244159 

By (193) By (194) 

q>= 40° 43' 48". 5 sin 9.814579 cot 10.064971 sec 0.120450 

A = 0°10' 9" tan 7.470148 sin 7.470146 

B = 0° 6' 37 ".3 tan 7.284727 sin 7.284727 
8' = 88° 41' 26".87 tan 11.340859 

t + B= 5° 36' 55" sin 8.990557 

.-. t= 5° 30' 18" 

= Oh. 22m. Is. 



124 TRIGONOMETRY. 

This method may be used by surveyors for finding the true 
meridian, and is useful when time enters the computation, for 
an ordinary time piece may be used. Thus, observe when 
Polaris is in the same vertical with e Ursae Ma j oris (the second 
star in the handle of the great dipper from the bowl) ; then in 
22 minutes the polar star will be on the meridian. 

The hour angle of Polaris when it and a Ursae Majoris have 
the same azimuth is 2h. 11m. 36s. 

When Polaris and S Ursae Majoris are in the same vertical 
the common azimuth is 0° 26' 28". 6 and the corresponding 
hour angle of Polaris is Oh. 37m. 10.45s. 

The hour angles of a and <5 Ursae Majoris will be much less 
than that of Polaris when the azimuths are equal. 



TRIGONOMETBIO FORMULA. 



125 







Solution of Oblique Triangles. 






A/ Ti XfJ 


Fig. 108. 




GIVEN. 


SOUGHT. 


FORMULAE. 


22 


A, B,a 


C,b,c 


fl _ i QftO / A I p\ 7, ^ cjr, Z> 


v n " sin ^t * 








i~ rfn f 4 I PI 


C ~ sin J. " in {A ' ^ 


23 


A, a, b 


B,C, c 


sin 5 =. Sm ^ . 6, C = 180° (^ + i?). 


~ sin A 


24 


C,a, b 


YzU + B) 


y 2 (A + B) = 90°-%C 


25 




\i (A - B) 


tany 2 (A-B)=~^tany 2 (A + B) 


26 




A, B 


A = y,(A + B)+}4(A-B\ 

b = y 2 (A + B) - y 2 u - B) 


27 




c 


c-(a\ m cos ^^+J5) _ , a b) sin^U + P) 
C - {a + b >cosyAA-B)- {a b >siny 2 (A-B) 


28 
29 

30 
31 


a, 6, c 


area 
A 


K = y, a b sin C. 


Let s = i^(a + & + c);sin^^ = j/<?=^i^ 


cos^ =4/ S( r ^;tan^ = j/ (s "^ 
"* y be y s (s— a) 




32 




area 


2 (.? - 6) (s - c) 

vers ^4 = — ~ 

be 


IZ ~ Vs (s — a) (s — b) (s — c) 


S3 


A, B, C, a 


area 


_ a 2 sin B . sin (7 
"~ 2 sin ^L 



126 TRIGONOMETRIC FORMULA. 



GENERAL FORMULA. 



1 



34 sin A — — v = V 1 — cos 2 4^ = tan A cos ^4 

cosec A 

35 sin ^4 = 2 sin ^ J. cos J^ ^4 = vers -4 cot ^ .4 



86 sin A = ^ ^ vers 2 4. = ^ (1 — cos 2 ^L) 



1 



37 cos A = 7 = V 1 — sin 2 4. = cot A sin .4 

sec A 

38 cos .4 = 1 — vers A = 2 cos 2 y%A—\ — 1 — 2 sin 2 ^ 41 



40 
41 
42 
43 
44 

45 

46 

47 
48 

49 
50 
51 
52 



cos 4. = cos 2 ¥ 2 A — sin 2 % A = \ / y 2 + y>cos2A 
1 sin 4^ 



tan 4. = — — — j = : = V sec 2 A— \ 

cot 4L cos 41 



y cos 2 



4/ 1 — cos 2 A _ ?i5L^_^L_ 

cos A "I -j- cos 2 ^4 



1 — cos 2 A vers 2,4 . . - , a 

tan -4 = — . _ ■- = — — zr— -r = exsec A cot >£ A 
sin 2 4. sm2i * 



1 cos A 



cot 4. b= j =■ -; 7 = 4/ cosec 2 A — \ 

tan 4. sm 4. 



sin 2 ^4 sm 2 4. 1 4- cos 2 A 

COt .4 = --„- a — — it - T — = c^l — 

1 — cos 2 A vers 2 A sin 2 4. 

tan ^2 ^4 

cot 4. = .- 

exsec A 

vers A — 1 — cos 4. = sin A tan y>A — 2 sin 2 ^ .4 

vers 4. = exsec A cos -4 

vers A 



exsec A — sec A — 1 = tan A tan Y^A = 



cos .4 



. , , . /I — cos 4. / vers 4. 

sm^^ = |/ — s — = y—r- 



sin 2 4. = 2 sin 4. cos 4. 



cos ^ -4 =» i/ — -L-g 



cos A 



cos 2 .4 = 2 cos 2 4 — 1 = cos 2 A — sin 2 A = 1 — 2 sin" A 



TRIGONOMETRIC FORMULA. 127 



General Formula. 



t-«« a. 1 , a tan A 

53. tan Vo A = .— . -r- = cosec 

'* 1 + sec A 



*a *. o a 2 tan A 

54. tan 2 A = ■. 



1 — COS^l i/l — COS.^1 

-4 — cot ud = ■ -. — 2 — " = 4/ i i „ ~ ^ 

sin -4 r 1 + cos 4 



1 — tan 2 ^ 

tK . , , , . sin A 1 + cos J. 1 

55. cot. }4A = — - — -. — - — = —--7 

vers.<4 sini cosec A — cot A 

56. cot 2 ^4 = — — — — 

2 cot A 

erf , . . % vers A 1 — cos .4 

57. vers \i A- ? ™ = - ■ 

1 + ^1 — Vz vers .4 2 + V2 (1 + cos A) 

58. vers 2 ^4 = 2 sin 2 A 
1 — cos ^4 



59. exsec \&A = 



0. exsec 2 .4 = 



(1 + cos A) + Va (l + cos ^) 
tan 2 ^ 



1 — tan 2 ^ 

61. sin {A ± B) = sin ^4 . cos B ± sin P . cos ^l 

62. cos ( A ± B) = cos A . cos £ T sin A . sin # 

63. sin ^4 + sin B = 2 sin %(A-\-B) cos ^ U — J5) 

64. sin .4 — sin B = 2 cos ^ (A -f 2?) sin ^ (^ — 5) 

65. cos A + cosB = 2cos}4(A + B) cos ^ (.4 - B) 

66. cos J5 — cos A = 2 sin \& (^ 4- J?) sin 14 (A — B) 

67. sin* .4 — sin 2 B = cos 2 P — cos 2 .4 = sin (A -\- B) sin {A — B) 

68. cos 2 A — sin 2 i? = cos {A + B) cos {A — B) 

sin (i + B) 



69. tan A -f tan B = 



70. tan ^4 — tan B = 



cos A . cos 5 

sin (A — P) 
cos ^4 . cos B 



EXPLANATION 

OF 

THE TABLES. 





cos 60° ' 


50° 


log sin 9.884254 


60° 


log cos 9.698970 


subtracting, 


0.185284 



The arithmetical complement of a number is the remainder 
after subtracting it from 10. It is often used to avoid the 
necessity of subtracting one logarithm from another. Instead 
of subtracting a logarithm, we may add its arithmetical com- 
plement and subtract 10 from the result. Thus a — b is the 
same as a + (10 — b) — 10. 

The arithmetical complement is indicated by the abbreviation 
" ar co," or "a. c." Thus to find the logarithm of 
sin 50° 



we have 



or, 

50° log sin 9.884254 

60° ar co log cos 0.301030 

adding, 10.185284 

subtracting 10 

gives 0.185284, 

which is the same as the preceding result. 

Tlie logarithm of a number consists of two parts, a 
whole number called the characteristic, and a decimal called 
the mantissa. All numbers which consist of the same figures 
standing in the same order have the same mantissa, regard- 
less of the position of the decimal point in the number, 
or of the number of ciphers which precede or follow the 
significant figures of the number. The value of the char- 
acteristic depends entirely on the position of the decimal point 
in the number. It is always one less than the number of 



130 TRIGONOMETRY. 

figures in the number to the left of the decimal point. The 
value is therefore diminished by one every time the decimal 
point of the number is removed one place to the left, and vice 
versa. Thus 



Number. 


Logarithm. 


3S40. 


4.141136 


1384.0 


3.141136 


138.40 


2.141136 


13.84 


1.141138 


1.384 


0.141136 


.1384 


1.141136 


.01384 


~2. 141136 


.001384 


"3.141136 


etc. 


etc. 



The mantissa is always positive even when the characteristic 
is negative. We may avoid the use of a negative characteristic 
by arbitrarily adding 10, which may be neglected at the close 
of the calculation. By this rule we have 

Number. Logarithm. 

1.384 0.141136 

.1384 9.141136 

.01384 8.141136 

.001384 7.141133 
etc. etc. 

TSTo confusion need arise from this method in finding a number 
from its logarithm; for although the logarithm 6.141136 repre- 
sents either the number 1,384,000, or the decimal .0001384, yet 
these are so diverse in their values that we can never be uncer- 
tain in a given problem which to adopt. 

The first table contains the mantissas of logarithms, car- 
ried to six places of decimals, for numbers between 1 and 9999, 
inclusive. The first three figures of a number are given in the 
first column, the fourth at the top of the other columns. The 
first two figures of the mantissa are given only in the second 
column, but these are understood to apply to the remaining 
four figures in either column following, which are comprised 
between the same horizontal lines with the two. 

If a number (after cutting off the ciphers at either end) con- 
sists of not more than four figures, the mantissa may be taken 
direct from the table; but-by interpolation the logarithm of a 
number having six figures may be obtained. The last column 
contains the average difference of consecutive logarithms on 



EXPLANATION OF TABLES. 131 

the same line, but for a given case the difference needs to be 
verified by actual subtraction, at least so far as the last figure 
is concerned. The lower part of the page contains a complete 
list of differences, with their multiples divided by 10. 

To find the logarithm of a number having six 
figures :— Take out the mantissa for the four superior places 
directly from the table, and find the difference between this 
mantissa and the next greater in the table. Add to the man- 
tissa taken out the quantity found in the table of proportional 
parts, opposite the difference, and in the column headed by the 
fifth figure of the number; also add -^ the quantity in the col- 
umn headed by the sixth figure. The sum is the mantissa 
required, to which must be prefixed a decimal point and the 
proper characteristic. 

Example. — Find the log of 23.4275. 

For 2342 mantissa is 369537 

" diff. 185 col. 7 129.5 

" " " " 5 9.2 



Arts. For 23.4275 log is 1.369726 
The decimals of the corrections are added together to deter- 
mine the nearest value of the sixth figure of the mantissa. 

To find the number corresponding to a given 
logarithm. — If the given mantissa is not in the table find the 
one next less, and take out the four figures corresponding to it ; 
divide the difference between the two mantissas by the tabu- 
lar difference in that part of the table, and annex the figures of 
the quotient to the four figures already taken out. Finally, 
place the decimal point according to the rule for characteristics, 
prefixing or annexing ciphers if necessary. The division re- 
quired is facilitated by the table of proportional parts, which 
furnishes by inspection the figures of the quotient. 

Example. — Find the number of which the logarithm is 

8.263927 8.263927 

First 4 figures 1836 from 263873 



Diff. 54.0 

Tabular diff. =236 .\ 5th fi<r. =2 47.2 



6.80 
6th fig. = 3 7.08 



Ans. No. ='.183,623,000. 



132 TRIGONOMETRY. 

The number derived from a six-place logarithm is not relia- 
ble beyond the sixth figure. 

At the end of the first table is a small table of logarithms of 
numbers from 1 to 100, with the characteristic prefixed, for 
easy reference when the given number does not exceed two 
digits. But the same mantissas may be found in the larger 
table. 

The logarithmic sine, tangent, etc. of an arc 

is the logarithm of the natural sine, tangent, etc. of the 
same arc, but with 10 added to the characteristic to avoid 
negatives. This table gives log sines, tangents, cosines, and 
cotangents for every minute of the quadrant. With the 
number of degrees at the left side of the page are to be read 
the minutes in the left-hand column ; with the degrees on 
the right-hand side are to be read the minutes in the right-hand 
column. When the degrees appear at the top of the page the 
top headings must be observed, when at the bottom those at 
the bottom. Since the values found for arcs in the first quad- 
rant are duplicated in the second, the degrees are given from 
0° to 180°. The differences in the logarithms due to a change 
of one second in the arc are given in adjoining columns. 

To find the log. sin, cos, tan, or cot of a given 
arc. : Take out from the proper column of the table the log- 
arithm corresponding to the given number of degrees and 
minutes. If there be any seconds multiply them by the ad- 
joining tabular difference, and apply their product as a cor- 
rection to the logarithm already taken out. The correction is 
to be added if the logarithms of the table are increasing with 
the angle, or subtracted if they are decreasing as the angle in- 
creases. In the first quadrant the log sines and tangents in- 
crease, and the log. cosines and cotangents decrease as the 
angle increases. 

Example.— Find the log sin of 9° 28' 20". 

Log sin of 9° 28' is 9.216097 

Add correction 20 X 12.62 252 

Ans. 9.216349 
Example.— Find the log cot of 9° 28' 20". 

Log cotan of 9° 28' is 10.777948 

Subtract correction 20 X 12.97 259 

Ans. 10 777689 



EXPLANATION OF TABLES. 133 

To find the angle or arc corresponding to a 
given logarithmic sine, tangent, cosine, or co- 
tangent.— If the given logarithm is found in the proper 
column take out the degrees and minutes directly ; if not, find 
the two consecutive logarithms between which the given 
logarithm would fall, and adopt that one which corresponds to 
the least number of minutes ; which minutes take out with the 
degrees, and divide the difference between this logarithm and 
the given one by the adjoining tabular difference for a quo- 
tient, which will be the required number of seconds. 

With logarithms to six places of decimals the quotient is 
not reliable beyond the tenth of a second. 

Example. — 9.383731 is the log tan of what angle? 
Next less 9.383682 gives 13° 36' 

Diff. ~T9.00 + 9.20 = 05".3 



Ans. 13° 36' 05".3 

Example. — 9.249348 is the log cos of what angle? 
JSi ext greater 583 gives 79' 46' 

Diff. 235 -T- 11.67 = 20M 



Ans. 79° 46' 20U 

The above rules do not apply to the first two pages of this 

table (except for the column headed cosine at top) because 

here the differences vary so rapidly that interpolation made by 

them in the usual way will not give exact results. 

On the first two pages, the first column contains the number 
of seconds for every minute from l'to2°; the minutes are 
given in the second, the log. sin. in the third, and in the fourth 
are the last three figures of a logarithm which is the difference 
between the log sin and the logarithm of the number of sec- 
onds in the first column. The first three figures and the char- 
acteristic of this logarithm are placed, once for all, at the head 
of the column. 

To find the log sin of an arc less than 2° given 

to seconds.— Reduce the given arc to seconds, and take the 
logarithm of the number of seconds from the table of loga- 
rithms, and add to this the logarithm from the fourth column 
opposite the same number of seconds. The sum is the log sin 
required. 

The logarithm in the fourth column may need a slight inter- 



J34 TRIGONOMETRY. 

polation of the last figure, to make it correspond closely to the 
given number of seconds. 

Example.— Fm& the log sin of 1° 39' 14". 4. 

1° 39' 14".4 = 5954".4 log 3.774833 

add (q — I) 4.685515 

Ans. log sin 8.460353 

Log tangents of small arcs are found in the same way, only 
taking the last four figures of {q — from the fifth column. 

Example.— Find the log tan of 0° 52' 35". 

52' 35" = (3120" + 35") = 3155" log 3.498999 

add (g-0 4.685609 

Ans. log tan 8.184603 

To find the log cotangent of an angle less than 
2° given to seconds,— Take from the column headed ( q-\- 1) 
the logarithm corresponding to the given angle, interpolating 
for the last figure if necessary, and from this subtract the loga- 
rithm of the number of seconds in the given angle. 

Example.— Find the log cotan of 1° 44' 22".5. 

q + I 15.314292 
6240" + 22",5 = 6262.5 log 3.796748 

Ans. 11.517544 

These two pages may be used in the same way when the 
given angle lies between 88° and 92°, or between 178° and 180°; 
but if the number of degrees be found at the bottom of the page, 
the title of each column will be found there also; and if the 
number of degrees be found on the right hand side of the page, 
the number of minutes must be found in the right hand col- 
umn, and since here the minutes increase upward, the number 
of seconds on the same line in the first column must be dimin- 
ished by the odd seconds in the given angle to obtain the num- 
ber whose logarithm Is to be used with {q ± I) taken from the 
table. 

Example.— Find the log cos of 88° 41' 12". 5 

(q - I) 4.685537 
4740" - 12".5 = 4727.5 log 3,674631 

Ans. 8.360168 



EXPLANATION OF TABLES. 135 

Example.— Find the log tan of 90° 30' 50'. 

q + l 15.314413 
1800" + 50" = 1850' log 3.267 172 

Ans. 12.047241 

To find the arc corresponding to a given log 
sin, cos, tan, or cotan which falls within the 
limits of the first two pages of the table. 

Find in the proper column two consecutive logarithms be- 
tween which the given logarithm falls. If the title of the 
given function is found at the top of that column read the 
degrees from the top of the page; if at the bottom read from 
the bottom. 

Find the value of (q — I) or (q -f- I), as the case may require, 
corresponding to the given log (interpolating for the last figure 
if necessary). Then if q = given log and I = log of number of 
seconds, n, in the required arc, we have at once I = q — (q — I) 
or I = (q -\-l) — q y whence n is easily found. 

Find in the first column two consecutive quantities between 
which the number n falls, and if the degrees are read from 
the left hand side of the page, adopt the less, take out the 
minutes from the second column, and take for the seconds 
the difference between the quantity adopted and the number 
n. But if the degrees are read from the right hand side of the 
page, adopt the greater quantity, take out the minutes on the 
same line from the right-hand column, and for the seconds 
take the difference between the number adopted and the num- 
ber 71. 

Example. — 11.734288 is the log cot of what arc? 

q + I 15.314376 

q 11 .734268 

.-. n- 3802.8 3.580103 

For 1° adopt 3780. giving 03' 

Difference 22". 8 

Am. 1° 03' 22". 8 or 178° 56' 37".2. 

Example. — 8.201795 is the log cos of what arc? 

q — I 4.685556 

q 8.201795 

.*. n= ■ 3282". 8 3.516239 

For 89° adopt 3300. giving 05' 

Difference 17". 2 

Ans. 89° 05' 17".2 or 90° 54' 42". 8. 



LOGARITHMS OF NUMBERS. 



LOGABITHMS OF NUMBERS. 



No. 100 L. 000.] 



[No. 109 L. 040. 



N. 





1 


2 


8 


4 I 


5 


6 


7 


8 


9 


Diff. 


100 000000 
1 4321 

9 ! ftA/Vl 


0434 
4751 
9026 


0868 
5181 
9451 


1301 
5609 
9876 


1734 i 

6038 


2166 
6466 


2598 
6894 


3029 
7321 


3461 
7748 


3891 
8174 


432 

426 






0900 i 

4521 
8700 


0724 

4940 
9116 

' 3252 

] 7350 


1147 
5360 
9532 

3664 
7757 


1570 

5779 
9947 

4075 
8164 


1993 
6197 


2415 
6616 


424 
420 


3 

4 


012837 
7033 


3259 
7451 


3680 
7868 


4100 

8284 


0361 
4486 
8571 


0775 
4896 
8978 


416 
412 
408 


5 

6 


021189 
5306 
9384 


1603 
5715 

9789 


2016 
6125 


2428 
6533 


2841 I 
6942 


1 


0195 
4227 

8223 


0600 
4628 
8620 


1004 : 

5029 
9017 j 


! 1408 
; 5430 
; 9414 


1812 
5830 
9811 


2216 
6230 


2619 
6629 


3021 
7028 


404 

400 


8 
9 


033424 

7426 
04 


3826 

7825 


0207 


0602 


0998 


397 



Proportional Parts. 



Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


434 


43.4 


86.8 


130.2 


173.6 


217.0 


260.4 


303.8 


347.2 


390.6 


433 


43.3 


86.6 


129.9 


173.2 


216.5 


259.8 


303.1 


346.4 


389.7 


432 


43.2 


86.4 


129.6 


172.8 


216.0 


259.2 


302.4 


345.6 


388.8 


431 


43.1 


86.2 


129.3 


172.4 


215.5 


258.6 


301.7 


344.8 


387.9 


430 


43.0 


86.0 


129.0 


172.0 


215.0 


258.0 


301.0 


344.0 


387.0 


429 ; 


42.9 


85.8 


128.7 


171.6 


214.5 


257.4 


300.3 


343.2 


386.1 


428 


42.8 


85.6 


128.4 


171.2 


214.0 


256.8 


299.6 


342.4 


385.2 


427 


42.7 


85.4 


128.1 


170.8 


213.5 


256.2 


298.9 


341.6 


384.3 


426 


42.6 


85.2 


127.8 


170.4 


213.0 


255.6 


298.2 


340.8 


383.4 


425 


42.5 


85.0 


127.5 


170.0 


212.5 


255.0 


297.5 


340.0 


382.5 


424 


42.4 


84.8 


127.2 


169.6 


212.0 


254.4 


296.8 


339.2 


381.6 


423 


42.3 


84.6 


126.9 


169.2 


211.5 


253.8 


296.1 


338.4 


380.7 


422 


42.2 


84.4 


126.6 


168.8 


211.0 


2.53.2 


295.4 


337.6 


379.8 


421 


42.1 


84.2 


126.3 


168.4 


210.5 


252.6 


294.7 


336.8 


378.9 


420 


42.0 


84.0 


126.0 


168.0 


210.0 


252.0 


294.0 


336.0 


378.0 


419 


41.9 


83.8 


125.7 


167.6 


209.5 


251.4 


293.3 


335.2 


377.1 


418 


41.8 


83.6 


125.4 


167.2 


209.0 


250.8 


292.6 


334.4 


376.2 


417 


41.7 


83.4 


125.1 


166.8 


208.5 


250.2 


291.9 


333.6 


375.3 


416 


41.6 


&3.2 


124.8 


166.4 


208.0 


249.6 


291.2 


332.8 


374.4 


415 


41.5 


83.0 


124.5 


166.0 


207.5 


249.0 


290.5 


332.0 


373.5 


414 


41.4 


82.8 


124.2 


165.6 


207.0 


248.4 


289.8 


331.2 


372.6 


413 


41.3 


82.6 


123.9 


165.2 


206.5 


247.8 


289.1 


330.4 


371.7 


412 


41.2 


82.4 


123.6 


164.8 


206.0 


247.2 


288.4 


329.6 


370.8 


411 


41.1 


82.2 


123.3 


164.4 


205.5 


246.6 


287.7 


328.8 


369.9 


410 


41.0 


82.0 


123.0 


164.0 


205.0 


246.0 


287.0 


328.0 


369.0 


409 


40.9 


81.8 


122.7 


163.6 


204.5 


245.4 


286.3 


327.2 


368.1 


408 


40.8 


81.6 


122.4 


163.2 


204.0 


244.8 


285.6 


326.4 


367.2 


407 


40.7 


81.4 


122.1 


162.8 


203.5 


244.2 


284.9 


325.6 


366.3 


406 


40.6 


81.2 


121.8 


162.4 


203.0 


243 6 


284.2 


324.8 


365.4 


405 


40.5 


81.0 


121.5 


162.0 


202.5 


243.0 


283.5 


324.0 


364.5 


404 


40.4 


80.8 


121.2 


161.6 


202.0 


242.4 


282.8 


323.2 


363.6 


403 


40.3 


80.6 


120.9 


161.2 


201.5 


241.8 


282.1 


322.4 


362.7 


402 


40.2 


80.4 


120.6 


160.8 


201.0 


241 2 


281.4 


321.6 


361.8 


401 


40.1 


80.2 


120.3 


160.4 


200.5 


240.6 


280.7 


320.8 


360.9 


400 


40.0 


80-0 


120.0 


160.0 


200.0 


240.0 


280.0 


320.0 


360.0 


399 


39.9 


79.8 


119.7 


159.6 


199.5 


239.4 


279.3 


319.2 


359.1 


398 


39.8 


79.6 


119.4 


159.2 


199.0 


238.8 


278.6 


318.4 


358.2 


397 


39.7 


79.4 


119.1 


158.8 


198.5 


238.2 


277.9 


317.6 


357.3 


396 


39.6 


79.2 


118.8 


158.4 


198.0 


237.6 


277.2 


316.8 


356.4 


395 


39.5 


79.0 


118.5 


158.0 


197.5 


237.0 


276.5 


316 


355.5 



139 



LOGARITHMS OF NUMBERS. 



No. 


110 L. 041.] 














[No 


. 119 L. 078. 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 


110 
1 
2 


041393 
5323 

9218 


1787 
5714 
9606 


2182 
6105 
9993 


2576 
6495 


2969 

6885 


&362 
7275 


3755 
7664 


4148 
8053 


4540 
8442 


4932 
8830 


393 
390 




0380 
4230 
8046 


0766 
4613 


1153 

4996 
8805 


1538 
5378 
9185 


1924 
5760 
9563 


2309 
6142 
9942 


2694 
6524 


386 
383 


3 

4 


053078 
6905 


3463 

7286 


3846 
7666 




0320 
4083 
7815 


379 
376 
373 


5 
6 

7 


060698 
4458 
8186 


1075 

4832 
8557 


1452 

5206 
8928 


1829 
5580 
9298 


2206 
5953 
9668 


2582 
6326 


2958 
6699 


3333 
7071 


3709 
7443 


0038 
3718 
7368 


0407 
4085 
7731 


0776 
4451 

8094 


1145 
4816 
$457 


1514 
5182 
8819 


370 
366 
363 


8 
9 


071882 
5547 


2250 
5912 


2617 
6276 


2985 
6640 


3352 
7004 



Proportional Parts. 



Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


395 


39.5 


79.0 


118.5 


158.0 


197.5 


237.0 


276.5 


316.0 


355.5 


394 


39.4 


78.8 


118.2 


157.6 


197 


236.4 


275.8 


315.2 


354.6 


393 


39.3 


78.6 


117.9 


157.2 


196.5 


235.8 


275.1 


314.4 


353.7 


392 


39.2 


78.4 


117.6 


156.8 


196.0 


235.2 


274.4 


313.6 


352.8 


391 


39.1 


78.2 


117.3 


156.4 


195.5 


234.6 


273.7 


312.8 


351.9 


390 


39.0 


78.0 


117.0 


156.0 


195.0 


234.0 


273.0 


312.0 


351.0 


389 


38.9 


77.8 


116.7 


155.6 


194.5 


233.4 


272.3 


311.2 


350.1 


388 


38.8 


77.6 


116.4 


155.2 


194.0 


232.8 


271.6 


310.4 


349.2 


387 


38.7 


77.4 


116.1 


154.8 


193.5 


232.2 


270.9 


309.6 


348.3 


386 


38.6 


77.2 


115.8 


154.4 


193.0 


231.6 


270.2 


308.8 


347.4 


385 


38.5 


77.0 


115.5 


154.0 


192.5 


231.0 


269.5 


308.0 


£46.5 


384 


38.4 


76.8 


115.2 


153.6 


192.0 


230.4 


268.8 


307.2 


345.6 


383 


38.3 


76.6 


114.9 


153.2 


191.5 


229.8 


268.1 


306.4 


344.7 


382 


38.2 


76.4 


114.6 


152.8 


191.0 


229.2 


267.4 


305.6 


343.8 


381 


38.1 


76.2 


114.3 


152.4 


190.5 


228.6 


266.7 


304.8 


£42.9 


380 


38.0 


76.0 


114.0 


152.0 


190.0 


228.0 


266.0 


304.0 


342.0 


379 


37.9 


75.8 


113.7 


151.6 


189.5 


227.4 


265.3 


303.2 


341.1 


378 


37.8 


75.6 


113.4 


151.2 


189.0 


226.8 


264.6 


302.4 


340.2 


377 


37.7 


75.4 


113.1 


150.8 


188.5 


226.2 


263.9 


301.6 


339.3 


376 


37.6 


75.2 


112.8 


150.4 


188.0 


225.6 


263.2 


300.8 


338.4 


375 


37.5 


75.0 


112.5 


150.0 


187.5 


225.0 


262.5 


300.0 


337.5 


374 


37.4 


74.8 


112.2 


149.6 


187.0 


224.4 


261.8 


299.2 


336.6 


373 


37.3 


74.6 


111.9 


149.2 


186.5 


223.8 


261.1 


298.4 


335.7 


372 


37.2 


74.4 


111.6 


148.8 


186.0 


223.2 


260.4 


297.6 


334.8 


371 


37.1 


74.2 


111.3 


148.4 


185.5 


222.6 


259.7 


296.8 


333.9 


370 


37.0 


74.0 


111.0 


148.0 


185.0 


222.0 


259.0 


296.0 


333.0 


369 


36.9 


73.8 


110.7 


147.6 


184.5 


221.4 


258.3 


295.2 


332.1 


368 


36.8 


73.6 


110.4 


147.2 


184.0 


220.8 


257.6 


294.4 


331.2 


367 


36.7 


73.4 


110.1 


146.8 


183.5 


220.2 


256.9 


293.6 


830.3 


366 


36.6 


73.2 


109.8 


146.4 


183.0 


219.6 


256.2 


292.8 


329.4 


S65 


36.5 


73.0 


109.5 


146.0 


182.5 


219.0 


255.7 


292.0 


328.5 


364 


36.4 


72.8 


109.2 


145.6 


182.0 


218.4 


254.8 


291.2 


327.6 


363 


36.3 


72.6 


108.9 


145.2 


181.5 


217.8 


254.1 


290.4 


326.7 


362 


36.2 


72.4 


108.6 


144.8 


181.0 


217.2 


253.4 


289.6 


325.8 


361 


36.1 


72.2 


108.3 


144.4 


180.5 


216.6 


252.7 


288.8 


324. S 


360 


36.0 


72.0 


108.0 


144.0 


180.0 


216.0 


252.0 


288.0 


324.0 


359 


35.9 


71.8 


107.7 


143.6 


179.5 


215.4 


251.3 


287.2 


323.1 


358 


35.8 


71.6 


107.4 


143.2 


179.0 


214.8 


250.6 


286.4 


322.2 


&57 


35.7 


71.4 


107.1 


142.8 


178.5 


214.2 


249.9 


285.6 


321.3 


356 


35.6 


71.2 


106.8 


142.4 


178.0 


213.6 


249.2 


284.8 


320.4 



140 



LOGABITHMS OF NUMBERS. 



No. 120 L. 070.] 



[No. 134 L. 130. 



N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 




079181 


9543 


9904 






1 












120 


0266 
3861 
7426 


0626 
4219 

7781 


0987 
4576 

8136 


1347 
4934 

8490 


1707 
5291 

8845 


2067 

5647 
9198 


2426 

6004 
9552 


1 360 


1 
2 
3 

4 
5 


082785 
6360 
9905 

093422 
6910 


3144 
6716 


3503 
7071 


357 
355 


0258 
3772 
7257 


0611 
4122 
7604 


0963 
4471 
7951 


1315 

4820 
8298 


1667 
5169 
8644 


2018 
5518 
8990 


2370 
5866 
9335 


2721 
6215 
9681 


3071 
6562 


352 
349 


0026 
3462 
6871 


346 

343 

341 


6 

7 
8 


100371 
3804 
7210 


0715 
4146 
7549 


1059 
4487 

7888 


1403 

4828 
8227 


1747 
5169 
8565 


1 2091 
5510 
8903 


2434 

5851 
9241 


2777 
6191 
9579 


3119 
6531 
9916 


0253 
3609 

6940 


338 
335 

333 


9 

130 
1 


110590 

3943 
7271 


0926 

4277 

7603 


1263 

4611 

7934 


1599 

4944 

8265 


1934 

5278 
8595 


| 2270 

5611 
8926 


2605 

5943 
9256 


2940 

6276 
9586 


3275 

6608 
9915 


0245 
3525 
6781 


330 
328 
325 


2 

3 

4 


120574 
3852 
7105 

13 


0903 
4178 
7429 


1231 
4504 
7753 


1560 
4830 
8076 


1888 
5156 
8399 


2216 

5481 
8722 


2544 

5806 
9045 


2871 
6131 
9368 


3198 
6456 
9690 


0012 


323 



Proportion al Parts. 



Diff. 


1 


2 


3 


4 


5 


355 


35.5 


71.0 


106.5 


142.0 


177.5 


354 


35.4 


70.8 


106.2 


141.6 


177.0 


353 


35.3 


70.6 


105.9 


141.2 


176.5 


352 


35.2 


70.4 


105.6 


140.8 


176.0 


351 


35.1 


70.2 


105.3 


140.4 


17'5.5 


350 


35.0 


70.0 


105.0 


140.0 


175.0 


349 


34.9 


69.8 


101.7 


139.6 


174.5 


348 


34.8 


69.6 


104.4 


139.2 


174.0 


347 


34.7 


69.4 


104.1 


138.8 


173.5 


346 


34.6 


69.2 


103.8 


138.4 


173.0 


345 


34.5 


69.0 


103.5 


138.0 


172.5 


344 


34.4 


68.8 


103.2 


137.6 


172.0 


343 


34.3 


68.6 


102.9 


137.2 


171.5 


342 


34.2 


68.4 


102.6 


136.8 


171.0 


341 


34.1 


68.2 


102.3 


136.4 


170.5 


340 


34.0 


68.0 


102.0 


136.0 


170.0 


339 


33.9 


67.8 


101.7 


135.6 


169.5 


338 


33.8 


67.6 


101.4 


135.2 


169.0 


337 


33.7 


67.4 


101.1 


134.8 


168.5 


336 


33.6 


67.2 


100.8 


134.4 


168.0 


335 


33.5 


67.0 


100.5 


134.0 


107.5 


334 


33.4 


66.8 


100.2 


133.6 


167.0 


333 


33.3 


66.6 


99.9 


133.2 


166.5 


332 


33.2 


66.4 


99.6 


132.8 


166.0 


331 


33.1 


66.2 


99.3 


132.4 


165.5 


330 


33.0 


66.0 


99.0 


132.0 


165.0 


329 


32.9 


65.8 


98.7 


131.6 


164.5 


328 


32.8 


65.6 


98.4 


131.2 


164.0 


327 


32.7 


65.4 


98.1 


130.8 


163.5 


326 


32.6 


65.2 


97.8 


130.4 


163.0 


325 


32.5 


65.0 


97.5 


130.0 


162.5 


324 


32.4 


64.8 


97.2 


129.6 


162.0 


323 


32.3 


64.6 


96.9 


129.2 


161.5 


322 


32.2 


64.4 


96.6 


128.8 


161.0 



6 


7 


8 


9 


213.0 


248.5 


284.0 


319.5 


212.4 


247.8 


283.2 


318.6 


211.8 


247.1 


282.4 


317.7 


211.2 


246.4 


281.6 


316.8 


210.6 


245.7 


280.8 


315.9 


210.0 


245.0 


280.0 


315.0 


209.4 


244.3 


279.2 


314.1 


208.8 


243.6 


278.4 


313.2 


208.2 


242.9 


277.6 


312.3 


207.6 


242.2 


276.8 


311.4 


207.0 


241.5 


276.0 


310.5 


206.4 


240.8 


275.2 


309.6 


205.8 


240.1 


274.4 


308.7 


205.2 


239.4 


273.6 


307.8 


204.6 


238.7 


272.8 


306.9 


204.0 


238.0 


272.0 


306.0 


203.4 


237.3 


271.2 


305.1 


202.8 


236.6 


270.4 


304.2 


202.2 


235.9 


269.6 


303.3 


201.6 


235.2 


268.8 


302.4 


201.0 


234.5 


268.0 


301.5 


200.4 


233.8 


267.2 


300.6 


199.8 


233.1 


266.4 


299.7 


199.2 


232.4 


265.6 


298.8 


198.6 


231.7 


264.8 


297.9 


198.0 


231.0 


264.0 


297.0 


197.4 


230.3 


263.2 


296.1 


196.8 


229.6 


262.4 


295.2 


196.2 


228.9 


261.6 


294.3 


195.6 


228.2 


260.8 


293.4 


195.0 


227.5 


260.0 


292.5 


194.4 


226.8 


259.2 


291.6 


193.8 


226.1 


258.4 


290.7 


193.2 


225.4 


257.6 


289.8 



141 



LOGARITHMS OF NgMBEES. 



No. 


135 L. 130.] 














[N 


3. 149 L. 175. 


N. 





1 


2 


8 


4 


5 


6 


7 


8 


9 


Diff. 


135 
6 

8 

9 

140 

1 


130334 
3539 
6721 

9879 

143015 

6128 
9219 


0655 
3858 
7037 


0977 
4177 
7354 


1298 
4496 
7671 


1619 
4814 
7987 


1939 
5133 
8303 


2260 
5451 

8618 


2580 
5769 
8934 


2900 
6086 
9249 


3219 
6403 
9564 


321 
318 
316 


0194 
3327 

6438 
9527 


0508 
3639 

6748 
9835 


0822 
3951 

7058 


1136 
4283 

7367 


1450 
4574 

7676 


1763 

4885 

7985 


2076 
5196 

8294 


2389 
5507 

8603 


2702 
5818 

8911 


314 
311 

309 


0142 
3205 
6246 
9266 


0449 
3510 
6549 
9567 


0756 
3815 
6852 
9868 


1063 
4120 
7154 


1370 
4424 

7457 


1676 
4728 
7759 


1982 
5032 
8061 


307 
305 
303 


2 
3 
4 


152288 
5336 
8362 


2594 
5640 
8664 


2900 
5943 
8965 


0168 
3161 
61:34 
9086 


0469 
3460 
6430 
9380 


0769 
3758 
6726 
9674 


1068 
4055 
7022 
9968 


301 
299 
297 
295 


5 
6 


161388 
4353 
7317 


1667 
4650 
7613 


1967 
4947 
7908 


2266 
5244 
8203 


25G4 
5541 
8497 


I 2863 

5838 

1 8792 


8 
9 


170262 
3186 


0555 

3478 


0848 
3769 


1141 
4060 


1434 
4351 


1726 
4641 


2019 
4932 


2311 
5222 


2603 
5512 


2895 
5302 


293 

291 



Proportional Parts. 



Diff. 


1 


2 


321 


32.1 


64.2 


320 


32.0 


64.0 


319 


31.9 


63.8 


318 


31.8 


63.6 


317 


31.7 


63.4 


316 


31.6 


63.2 


315 


31.5 


63.0 


314 


31.4 


62.8 


313 


31.3 


62.6 


312 


31.2 


62.4 


311 


81.1 


62.2 


310 


31.0 


62.0 


309 


30.9 


61.8 


308 


30.8 


61.6 


307 


30.7 


61.4 


306 


30.6 


61.2 


305 


30.5 


61.0 


304 


30.4 


60.8 


303 


30.3 


60.6 


302 


30.2 


60.4 


301 


30.1 


60.2 


300 


30.0 


60.0 


299 


29.9 


59.8 


293 


29.8 


59.6 


297 


29.7 


59.4 


296 


29.6 


59.2 


2:55 


29.5 


59.0 


204 


29.4 


58.8 


293 


29.3 


58.6 


292 


29.2 


58.4 


291 


29.1 


58.2 


290 


29.0 


58.0 


289 


28.9 


57.8 


288 


28.8 


57.6 


287 


28.7 


57.4 


286 


28.6 


57.2 



3 


4 


5 


6 


7 


8 


9 


96.3 


128.4 


160.5 


192.6 


224.7 


256.8 


288.9 


96.0 


128.0 


160.0 


192.0 


224.0 


256.0 


288.0 


95.7 


127.6 


159.5 


191.4 


223.3 


255.2 


287.1 


95.4 


127.2 


159.0 


190.8 


222.6 


254.4 


286.2 


95.1 


126.8 


158.5 


190.2 


221.9 


253.6 


285.3 


94.8 


126.4 


158.0 


189.6 


221.2 


252.8 


284.4 


94.5 


126.0 


157.5 


189.0 


220.5 


252.0 


283.5 


94.2 


125.6 


157.0 


188.4 


219.8 


251.2 


282.6 


93.9 


125.2 


156.5 


187.8 


219.1 


250.4 


281.7 


93.6 


124.8 


156.0 


187.2 


218.4 


249.6 


280.8 


93.3 


124.4 


155.5 


186.6 


217.7 


248.8 


279.9 


93.0 


124.0 


155.0 


186.0 


217.0 


248.0 


279.0 


92.7 


123.6 


154.5 


185.4 


216.3 


247.2 


278.1 


92.4 


123.2 


154.0 


184.8 


215.6 


246.4 


277.2 


92.1 


122.8 


153.5 


184.2 


214.9 


245.6 


276.3 


91.8 


122.4 


153.0 


183.6 


214.2 


244.8 


275.4 


91.5 


122.0 


152.5 


183.0 


213.5 


244.0 


274,5 


91.2 


121.6 


152.0 


182.4 


212.8 


243.2 


273.6 


90.9 


121.2 


151.5 


181.8 


212.1 


242.4 


272.7 


90.6 


120.8 


151.0 


131.2 


211.4 


241.6 


271.8 


90.3 


120.4 


150.5 


180.6 


210.7 


240.8 


270.9 


90.0 


120.0 


150.0 


180.0 


210.0 


240.0 


270.0 


89.7 


119.6 


149.5 


179.4 


209.3 


239.2 


269.1 


89.4 


119.2 


149.0 


178.8 


208.6 


238.4 


268.2 


89.1 


118.8 


148.5 


178.2 


207.9 


237.6 


267.3 


88.8 


118.4 


148.0 


177.6 


207.2 


236.8 


266.4 


88.5 


118.0 


147.5 


177.0 


206.5 


236.0 


265.5 


88.2 


117.6 


147.0 


176.4 


205.8 


235.2 


264.6 


87.9 


117.2 


146.5 


175.8 


205.1 


234.4 


263.7 


87.6 


116.8 


146.0 


175.2 


294.4 


233.6 


262.8 


87.3 


116.4 


145.5 


174.6 


203.7 


232.8 


261.9 


87.0 


116.0 


145.0 


174.0 


203.0 


232.0 


261.0 


86.7 


115.6 


144.5 


173.4 


202.3 


231.2 


260.1 


86.4 


115.2 


144.0 


172.8 


201.6 


230.4 


259.2 


86.1 


114.8 


143.5 


172.2 


200.9 


229.6 


258.3 


85.8 


114.4 


143.0 


171.6 


200.2 


228.8 


257.4 



142 



LOGARITHMS OF NUMBERS. 



No. 150 L. 176.] 



[No. 169 L. 230. 



150 


176091 


6381 6U70 


6959 


7248 


1 


S977 


9261 , 9552 


9839 


0126 

2985 


2 


181844 


2129 2415 


2700 


3 


4691 


4975 5259 


5542 


5825 


4 


7521 


7808 8084 


8366 


8647 


8 


190.332 


0612 


0892 


1171 


1451 


6 


3125 


3403 


3681 


3959 


4237 


7 


5900 


6176 


6453 


6729 


7005 


8 


8657 


8932 


9206 


9481 


9755 



7536 



0113 
3270 
6108 



1730 
4514 
7281 



7825 | 8113 



0699 
3555 
6391 



2010 
4792 
7556 



3839 
6674 
9190 



2289 
5069 

7832 



201397 j 1670 

4120 4391 
6826 7096 
9515 j 9783 



1943 2216 2488 



4663 4934 
7365 7634 



5204 
7904 



0029 0303 j 0577 

2761 | 3033 ; 3305 

5475 j 5746 I 6016 

8173 8441 I 8710 



212188 2454 
4814 5109 
7484 7747 



0051 0319 
2720 



5373 
8010 



5638 
8273 



0.586 
3252 
5902 
8536 



220108 


0370 


2716 


2976 


5309 


5568 


7887 


8144 


23 





0631 j 0892 1153 

3236 3496 3755 

5826 I 6084 . 6342 

8100 i 8657 8913 



0853 


1121 


1388 


1654 


1921 


3518 


3783 


4049 


4314 


4579 


6166 


6430 


6694 


6957 


7221 


8798 


9060 


9323 


9585 


9846 



1414 
4015 
6600 
9170 



1675 
4274 
6858 
9426 



4533 
7115 

9682 



8401 8689 



1272 1558 
4123 ! 4407 
6956 7239 
9771 



0051 
2567 I 2816 
5346 i 5623 



0850 
3577 



1124 

3848 

6556 
9247 



2196 2456 
4792 5051 
7372 7030 

9938 

0193 



Diff. 



289 

287 
285 
283 

281 

279 
278 
276 

274 

272 

271 



267 
266 
264 
262 

261 
259 
258 

256 



Proportional Parts. 



Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


885 


28.5 


57.0 


85.5 


114.0 


142.5 


171.0 


199.5 


228.0 


256.5 


281 


28.4 


56.8 


85.2 


113.6 


142.0 


170.4 


198.8 


227.2 


255.6 


283 


28.3 


56.6 


84.9 


113.2 


141 5 


169.8 


198.1 


226.4 


254.7 


282 


28.2 


56.4 


84.6 


112.8 


141.0 


169.2 


197.4 


225.6 


253.8 


281 


28.1 


56.2 


84.3 


112 4 


140.5 


168.6 


196.7 


224.8 


252.9 


280 


28.0 


56.0 


84.0 


112.0 


140.0 


168.0 


196.0 


224.0 


252.0 


279 


27.9 


55.8 


83.7 


111.6 


139.5 


167.4 


195.3 


223.2 


251.1 


278 


27.8 


55.6 


83.4 


111.2 


139.0 


166.8 


194.6 


222.4 


250.2 


277 


27.7 


55.4 


88.1 


110.8 


138.5 


166.2 


193.9 


221.6 


249.3 


276 


27.6 


55.2 


82.8 


110.4 


138.0- 


165.6 


193.2 


220.8 


248.4 


275 


27.5 


55.0 


82.5 


110.0 


137.5 


165.0 


192.5 


220.0 


247.5 


274 


27.4 


54.8 


82.2 


109.6 


137.0 


164.4 


191.8 


219.2 


246.6 


273 


27.3 


54.6 


81.9 


109.2 


136.5 


163.8 


191.1 


218.4 


245.7 


272 


27.2 


54.4 


81.6 


108.8 


136.0 


163.2 


190.4 


217.6 


244.8 


271 


27.1 


54.2 


81.3 


108.4 


135.5 


162.6 


189.7 


216.8 


243.9 


270 


27.0 


51.0 


81.0 


108.0 


135.0 


162.0 


189.0 


216.0 


243.0 


269 


26.9 


53.8 


80.7 


107.6 


134.5 


161.4 


188.3 


215.2 


242.1 


268 


26.8 


53.6 


80.4 


107.2 


134.0 


160.8 


187.6 


214.4 


241.2 


267 


26.7 


53.4 


80.1 


106.8 


133.5 


.160.2 


186.9 


213.6 


240.3 


266 


26.6 


53.2 


79.8 


106.4 


133.0 


159.6 


186.2 


212.8 


239.4 


265 


26.5 


53.0 


79.5 


106.0 


132.5 


159.0 


185.5 


212.0 


238.5 


264 


26.4 


52.8 


79.2 


105.6 


132.0 


158.4 


184.8 


211.2 


237.6 


263 


26.3 


52.6 


78.9 


105.2 


131.5 


157.8 


184.1 


210.4 


236.7 


262 


26.2 


52.4 


78.6 


104.8 


131.0 


157.2 


183.4 


209.6 


235.8 


261 


26.1 


52.2 


78.3 


104.4 


130.5 


156.6 


182.7 


208.8 


234.9 


260 


26.0 


52.0 


78.0 


104.0 


130.0 


156.0 


182.0 


208.0 


234.0 


259 


25.9 


51.8 


77.7 


103.6 


129.5 


155.4 


181.3 


207.2 


233.1 


258 


25.8 


51.6 


77.4 


103.2 


129.0 


154.8 


180.6 


206.4 


232.2 


257 


25.7 


51.4 


17.1 


102.8 


128.5 


154.2 


179.9 


205.6 


231.3 


256 


25.6 


51.2 


76.8 


102.4 


128.0 


153.6 


179.2 


204.8 


230,4 


255 


25.5 


51.0 


76.5 


102.0 


127.5 


153.0 


178.5 


204.0 


229.5 



143 



LOGARITHMS OF NUMBERS. 



No. 


170 L. 230.] 














[No. 189 L. 278. 


N. 





1 


2 


3 


4 . 


6 


6 


7 


8 


9 


Dm. 


170 
1 
2 
3 


230449 
2996 
5528 
8046 


0704 
3250 
5781 
8297 


0960 
3504 
6033 
8548 


1215 
3757 

6285 
8799 


1470 
4011 
6537 
9049 


1724 
4264 
6789 
9299 


1979 
4517 
7041 
9550 


2234 
4770 

7292 
9800 


2488 
5023 
7544 


2742 
5276 
7795 


255 
253 
252 


0050 
2541 
5019 

7482 
9932 


0300 
2790 
5266 
7728 


250 
243 

248 
£46 


4 
5 
6 

7 


240549 
3038 
5513 

7973 


0799 
3286 
5759 
8219 


1048 
3534 
6006 
&464 


1297 
3782 
6252 
8709 


1546 , 
4030 
6499 
8954 


1795 
4277 
6745 
9198 


2044 
4525 
6991 
9443 


2293 

4772 
7237 
9687 


0176 
2610 
5031 

7439 
9833 


245 
243 
£42 

241 
239 


8 
9 

180 
1 


250420 
2853 

5273 

7679 


0664 
3096 

5514 
7918 


0908 
3338 

5755 
8158 


1151 

3580 

5996 
8398 


1395 I 

3822 ! 

6237 

8637 


1638 
4064 

6477 

8877 


1881 
4306 

6718 
9116 


2125 
4548 

6958 
9355 


2368 
4790 

7198 
9594 


2 
3 
4 
5 
6 


260071 
2451 
4818 
7172 
9513 


0310 
2688 
5054 
7406 
9746 


0548 
2925 
5290 
7641 
9980 


0787 
3162 
5525 

7875 


1025 
3399 

5761 
8110 1 


1263 
3636 
5996 
8344 


1501 
3873 
6232 

8578 


1739 
4109 
6467 

8812 


1976 
4346 
6702 
9046 


2214 
4582 
6937 
9279 


233 
237 
235 
234 


0213 
2538 
4850 
7151 


0446 
2770 
5081 
7380 


0679 
3001 
5311 

7609 


0912 
3233 
5542 

7838 


1144 
3464 
5772 

8067 


1377 
3696 
6002 
8296 


1609 
3927 
6232 

8525 


233 

232 
230 
229 


7 
8 
9 


271842 
4158 
6462 


2074 
4389 
6692 


2306 
4620 
6921 



Proportional Parts. 



Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


255 


25.5 


51.0 


76.5 


102.0 


127.5 


153.0 


178.5 


204.0 


229.5 


254 


25.4 


50.8 


76.2 


101.6 


127.0 


152.4 


177.8 


203.2 


228.6 


253 


25.3 


50.6 


75.9 


101.2 


126.5 


151.8 


177.1 


202.4 


227.7 


252 


25.2 


50.4 


75.6 


100.8 


126.0 


151.2 


176.4 


201.6 


226.8 


251 


25.1 


50.2 


75.3 


100.4 


125.5 


150.6 


175.7 


200.8 


225.9 


250 


25 


50.0 


75.0 


100.0 


125.0 


150.0 


175.0 


200.0 


225.0 


249 


24.9 


49.8 


74.7 


99.6 


124.5 


149.4 


174.3 


199.2 


224.1 


248 


24.8 


49.6 


74.4 


99.2 


124.0 


148.8 


173.6 


198.4 


223.2 


247 


24.7 


49.4 


74.1 


98.8 


123.5 


148.2 


172.9 


197.6 


222.3 


246 


24.6 


49.2 


73.8 


98.4 


123.0 


147.6 


172.2 


196.8 


221.4 


245 


24.5 


49.0 


73.5 


98.0 


122.5 


147.0 


171.5 


196.0 


220.5 


244 


24.4 


48.8 


73.2 


97.6 


122.0 


146.4 


170.8 


195.2 


219.0 


243 


24.3 


48.6 


72.9 


97.2 


121.5 


145.8 


170.1 


194.4 


218.7 


242 


24.2 


48.4 


72.6 


96.8 


121.0 


145.2 


109.4 


193.6 


217.8 


241 


24.1 


48.2 


72.3 


96.4 


120.5 


144.6 


168.7 


192.8 


216.9 


240 


24.0 


48.0 


72.0 


96.0 


120.0 


144.0 


168.0 


192.0 


216.0 


239 


23.9 


47.8 


71.7 


95.6 


119.5 


143.4 


167.3 


191.2 


215.1 


238 


23.8 


47.6 


71.4 


95.2 


119.0 


142.8 


166.6 


190.4 


214.2 


237 


23.7 


47.4 


71.1 


94.8 


118.5 


142.2 


165.9 


189.6 


213.3 


236 


23.6 


47.2 


70.8 


&4.4 


118.0 


141.6 


165.2 


188.8 


212.4 


235 


23.5 


47.0 


70.5 


94.0 


117.5 


141.0 


164.5 


188.0 


211.5 


234 


23.4 


46.8 


70.2 


93.6 


117.0 


140.4 


163.8 


187.2 


210.6 


233 


23.3 


46.6 


69.9 


93.2 


116.5 


139.8 


163.1 


186.4 


209.7 


232 


23.2 


4G.4 


69.6 


92.8 


116.0 


139.2 


162.4 


185.6 


208.8 


231 


23.1 


46.2 


69.3 


92.4 


115.5 


138.6 


161.7 


184.8 


207.9 


230 


23.0 


46.0 


69.0 


92.0 


115.0 


138.0 


161.0 


184.0 


207.0 


229 


22 9 


45.8 


68.7 


91.6 


114.5 


137.4 


160.3 


183.2 


206.1 


228 


22.8 


45.6 


68.4 


91.2 


114.0 


136.8 


159.6 


182.4 


205 2 


227 


22.7 


45.4 


68.1 


90.8 


113.5 


136.2 


158.9 


181.6 


204.3 


226 


22.6 


45.2 


67.8 


90.4 


113.0 


135.6 


158 2 


180.8 


203.4 



144 



LOGABITHMS OF JSTUMBEB8. 



No. 


190 L. 278.] 














[No. 214 L. 332. 


N. 





1 


2 


3 


4 


5 


6 


7 


8 j 9 


Diff. 




278754 


8982 


9211 


9439 


9667 


9895 






1 




190 


0123 

2896 
4656 
6905 
9143 


0351 

2622 
4882 
7130 
9366 


0578 
2&49 
5107 
7354 
9589 


0806 
3075 
5332 
7578 
9812 


228 
227 
226 
225 
223 


1 
2 
3 

4 


281033 
3301 
5557 

7802 


1261 
3527 

5782 
8026 


1488 
3753 
6007 
8249 


1715 
3979 
6232 
8473 


1942 
4205 
6456 
8696 


2169 
4431 
6681 
8920 


5 
6 

7 
8 
9 


290035 
2256 
4466 
6665 
8853 


0257 
2478 
4687 
6884 
9071 


0480 
2699 
4907 
7104 
9289 


0702 
2920 
5127 
7323 
9507 


0925 
3141 
5347 
7542 
9725 


1147 
8333 
5567 
7761 
9943 


1369 

3584 
5787 
7979 


1591 
3804 
6007 
8198 


1813 
4025 
6226 
8416 


2034 
4246 
6446 
8635 


222 

221 
220 
219 


0161 

2331 
4491 
6639 

8778 


0378 

2547 
4706 
6854 

8991 


0595 

2764 
4921 
7068 
9204 


0813 

2980 
5136 
7282 
9417 


218 

217 
£16 
215 
213 


200 
1 
2 
3 
4 


301030 
3196 
5351 
7496 
9630 


1247 
3412 
5566 
7710 
9843 


1464 
3628 
5781 
7924 


1681 
3844 
5996 

8137 


1898 
4059 
6211 
8351 


1 2114 
: 4275 

i 6425 
8564 


0056 
2177 
4289 
6390 
S481 


0268 
2389 
4499 
6599 
8689 


04S1 
2600 
4710 
6809 
8898 


| 0693 

! 2812 

■' 4920 

7018 

9106 


0906 
3023 
5130 
7227 
9314 


1118 

3234 
5340 
7436 
9522 


1330 
3445 
5551 
7646 
8730 


1542 
3656 
5760 
7854 
9933 


212 
211 
210 
209 
208 


5 
6 

7' 
8 


311754 
3867 
5970 
8063 


1966 
4078 
6180 

8272 


9 

210 
1 
2 
3 


320146 

2219 

4282 
6336 
0380 


0354 

2426 
4488 
6541 
8583 


0562 

2633 
4694 
6745 

8787 


0769 

2839 
4899 
6950 
8991 


0977 

3046 
5105 
7155 
9194 


1184 

3252 
5310 
7359 
9398 


•1391 

3458 
5516 
7563 
9601 


1598 

3665 
5721 

7767 
9805 


1805 

3871 
5926 
7972 


2012 

4077 
6131 
8176 


207 

206 
205 
204 


0008 
2034 


0211 
2236 


203 

202 


4 


330414 


C617 


0819 


1052 : 1225 


i 1427 


1630 


1832 



Proportional Parts. 



Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


225 


22.5 


45.0 


67.5 


90.0 


112.5 


135.0 


157.5 


180.0 


202.5 


224 


22.4 


44.8 


67.2 


89.6 


112.0 


134.4 


156.8 


179.2 


201.6 


223 


22.3 


44.6 


66.9 


89.2 


111.5 


133.8 


156.1 


178.4 


200.7 


222 


22.2 


44.4 


66.6 


88.8 


111.0 


133.2 


155.4 


177.6 


199.8 


221 


22.1 


44.2 


66.3 


88.4 


110.5 


132.6 


154.7 


176.8 


198.9 


220 


22.0 


44.0 


66.0 


88.0 


110.0 


132.0 


154.0 


176.0 


198.0 


219 


21.9 


43.8 


65.7 


87.6 


109.5 


131.4 


153.3 


175.2 


197.1 


218 


21.8 


43.6 


65.4 


87.2 


109.0 


130.8 


152.6 


174.4 


196.2 


217 


21.7 


43.4 


65.1 


86.8 


108.5 


130.2 


151.9 


173.6 


195.3 


216 


21.6 


43.2 


64.8 


86.4 


108.0 


129.6 


151.2 


172.8 


194.4 


215 


21.5 


43.0 


64.5 


86.0 


107.5 


129.0 


150.5 


172.0 


193.5 


214 


21.4 


42.8 


64.2 


85.6 


107.0 


128.4 


149.8 


171.2 


192.6 


213 


21.3 


42.6 


63.9 


85.2 


106.5 


127.8 


149.1 


170.4 


191.7 


212 


21.2 


42.4 


63.6 


84.8 


106.0 


127.2 


148.4 


169.6 


190.8 


211 


21.1 


42.2 


63.3 


84.4 


105.5 


126.6 


147.7 


168.8 


189.9 


210 


21.0 


42.0 


63.0 


81.0 


105.0 


126.0 


147.0 


168.0 


189.0 


209 


20.9 


41.8 


62.7 


83.6 


104.5 


125.4 


146.3 


167.2 


188.1 


208 


20.8 


41.6 


62 4 


83.2 


104.0 


124.8 


145.6 


166 4 


187.2 


207 


20.7 


41.4 


62.1 


82.8 


103.5 


124.2 


144.9 


165.6 


186.3 


206 


20.6 


41.2 


61.8 


82.4 


103.0 


123.6 


144.2 


164.8 


185.4 


205 


20.5 


4d.O 


61.5 


82.0 


102.5 


123.0 


143.5 


164.0 


184.5 


204 


20.4 


40.8 


61.2 


81.6 


102.0 


122.4 


142.8 


163.2 


183.6 


203 


20.3 


40.6 


60.9 


81.2 


101.5 


121.8 


142.1 


162.4 


182.7 


202 


20.2 


40.4 


60.6 


80.8 


101.0 


121.2 


141.4 


161.6 


181.8 



145 



LOGARITHMS OF NUMBERS. 



No. 215 L. 332.] 



[No. 239 L. 380. 



N. 



215 
6 

7 



220 
1 
2 
3 

4 
5 
6 

7 

9 

230 
1 
2 
3 
4 



332438 
4454 
6460 
8456 



340444 

2423 
4392 
6353 
8305 



350248 
2183 
4108 
6026 
7935 
9835 



361728 
3612 
5488 
7356 
9216 

371063 
291-3 
4743 
6577 



2640 
4655 
6660 
8656 



0642 

2620 
4589 
6549 
8500 



0442 
2375 
4301 
6217 

8125 



0025 

1917 
3800 
5675 
7542 
9401 



1253 

3096 
4932 
6759 
8580 



2842 
4856 
6860 
8855 



0841 

2817 
4785 
6744 
8694 



0636 

2568 
4493 

6408 
8316 



0215 

2105 
3988 
5862 
7729 
9587 



1437 
8280 
5115 
6942 
8761 



3044 

50£7 
7060 
9054 



1039 

3014 
4981 
6939 



3246 
5257 
7260 
9253 



1237 

3212 
5178 
7135 
9083 



0829 
2761 
4685 
6599 
8506 



0404 

2294 
4176 
6049 
7915 
9?72 



1622 
3464 
5298 
7124 



1023 
2954 
4876 
6790 

8696 



0593 

2482 
4b63 
6236 
8101 
9958 



1806 
3647 
5481 
7306 
9124 



3447 
5458 
7459 
9451 



1435 

3409 
5374 

7330 
9278 



3649 

5658 
7659 
9650 



1632 

3606 
5570 
7525 
9472 



1216 
3147 

5068 
6981 

8886 



0783 

2671 
4551 
6423 

8287 



1410 
3339 
5260 
7172 
9076 



0972 

2859 
4739 
6610 
8473 



0143 
1991 
3831 
5664 
7488 
9306 



0328 
2175 
4015 
5846 
7670 
9487 



8 



3850 
5859 

7858 
9849 



1830 

3802 
5766 
7720 
9666 



1603 
3532 

5452 
7363 
9266 



1161 

3048 
4926 
6796 
8659 



0513 
2360 
4198 
6029 
7852 
9668 



4051 
6059 
8058 



0047 
2028 
3999 
5962 
7915 
9860 



1796 
3724 

5643 
7554 
9456 



1350 

3236 
5113 

6983 

8845 



Diff. 



4253 
6260 

8257 



0246 
2225 

4196 
6157 
8110 



0054 



5834 
7744 
9646 



1539 

3424 
5301 
7169 
9030 



0698 
2544 
4382 
6212 
8034 
9849 



2728 
4565 
6394 
8216 



0030 



Proportional Parts. 



Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


202 


20.2 


40.4 


60.6 


80.8 


101.0 


121.2 


141.4 


161.6 


181.8 


201 


20.1 


40.2 


60.3 


80.4 


100.5 


120.6 


140.7 


160.8 


180.9 


200 


20.0 


40.0 


60.0 


80.0 


100.0 


120.0 


140.0 


160.0 


180.0 


199 


19.9 


39.8 


59.7 


79.6 


99.5 


119.4 


139.3 


159.2 


179.1 


198 


19.8 


39.6 


59.4 


79.2 


99.0 


118.8 


138.6 


158.4 


178.2 


197 


19.7 


39.4 


59.1 


78.8 


98.5 


118.2 


137.9 


157.6 


177.3 


196 


19.6 


39.2 


58.8 


78.4 


98.0 


117.6 


137.2 


156.8 


176.4 


195 


19.5 


39.0 


58.5 


78.0 


97.5 


117.0 


136.5 


156.0 


175.5 


194 


19.4 


38.8 


58.2 


77.6 


97.0 


116.4 


135.8 


155.2 


174.6 


193 


19.3 


38.6 


57.9 


77.2 


96.5 


115.8 


135.1 


154.4 


173.7 


192 


19.2 


38.4 


57.6 


76. a 


96.0 


115.2 


134.4 


153.6 


172.8 


191 


19.1 


38.2 


57.3 


76.4 


95.5 


114.6 


133.7 


152.8 


171.9 


190 


19.0 


38.0 


57.0 


76.0 


95.0 


114.0 


133.0 


152.0 


171.0 


189 


18.9 


37.8 


56.7 


75.6 


94.5 


113.4 


132.3 


151.2 


170.1 


188 


18 8 


37.6 


56.4 


75.2 


94.0 


112.8 


131.6 


150.4 


169.2 


187 


18.7 


37 4 


56.1 


74.8 


93.5 


112.2 


130.9 


149.6 


168.3 


186 


18.6 


37.2 


55.8 


74.4 


93.0 


111.6 


130.2 


148.8 


167.4 


185 


18.5 


37.0 


55.5 


74.0 


92.5 


111.0 


129.5 


148.0 


166.5 


184 


18.4 


36.8 


55.2 


73.6 


92.0 


110.4 


128.8 


147.2 


165.6 


163 


18.3 


36.6 


54.9 


73.2 


91.5 


109.8 


128.1 


146.4 


104. 7 


182 


18.2 


36.4 


54.6 


72.8 


91.0 


109.2 


127.4 


145.6 


163.8 


181 


18.1 


36.2 


54.3 


72.4 


90.5 


108.6 


126.7 


144.8 


162.9 


180 


18.0 


36.0 


54.0 


72.0 


90.0 


108.0 


126.0 


144.0 


162.0 


179 


17.9 


35.8 


53.7 


71.6 


89.5 


107.4 


125.3 


143.2 


161.1 



14G 



LOGARITHMS OF NUMBERS. 



No. 240 L. 380.] 














[N 


0. 269 L. 431. 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 


240 


380211 


0392 


0573 


0754 


0934 


1115 


1296 


1476 


1656 


1837 


181 


1 


2017 


2197 


2377 


2557 


2737 


2917 


3097 


3277 


3450 3636 


180 


9, 


3S15 


3995 


4174 


4:353 


4533 


1 4712 


4891 


5070 5249 1 5423 


179 


a 


5606 


57S5 


5964 


6142 


6321 


! 6499 


6677 


6856 I 7034 7212 


173 


4 


7390 


7563 


7746 


7924 


8101 


8279 


8456 


86-34 8811 S 8989 


173 


5 


9166 


9343 


o=;oa 


9698 


9875 












j 0051 
! 1817 


0223 
1993 


0405 0582 
2169 2345 


0759 
2521 


177 


ft 


390935 111-2 1288 


1464 


1641 


176 


7 


2697 


2373 


3048 


3224 


3400 


3575 


3751 


3926 i 4101 


4277 


176 


8 


4452 


4627 


4802 


4977 


5152 


5326 


5501 


5676 5850 


6025 


175 


9 


6199 


6374 


6548 


6722 


6896 


7071 


7245 


7419 ' 7592 


7766 


174 


250 


7940 8114 


8287 


8461 


86-34 


1 8808 


8981 


9154 . 9323 


9501 


173 


1 


9674 9847 
















nnon 


0192 
1917 


0365 
20S9 


! 0538 
2261 


0711 
2433 


0883 ; 1056 1223 
2605 2777 ; 2949 


173 


2 


401401 1573 1 1745 


172 


a 


3121 | 3292 S464 


3635 


.3807 


3978 


4149 


4320 4492 4663 


171 


4 


4834 5005 5176 


5:346 


5517 


5683 


5858 


6029 6199 6370 


171 


5 


6540 6710 | 6881 


7051 


7221 


! 7391 


7561 


7731 7901 8070 


170 


6 


8240 ! 8410 


8579 


8749 


8918 


9087 


9257 


9426 9595 9764 


169 


7 


9933 
















0102 


0271 


0440 


0609 


0777 


0946 


1114 1233 1 1451 


169 


8 


411620 1788 


1956 


2124 


2293 


2461 


2629 


2796 2964 3132 


168 


9 


3300 3467 


3635 


3803 


3970 


4137 


4305 


4472 j 4639 4806 


167 


260 


4973 5140 


5307 


5474 


5641 


5808 


5974 


6141 6308 6474 


167 


1 


6641 | 6807 


6973 


7139 


7306 


| 7472 


7638 


7804 7970 8135 


166 


2 


8301 8467 


8633 


8798 


8964 


9129 


9295 


9460 9625 9791 


165 


3 


9956 














01 21 


0286 

1933 


0451 


0616 


0781 


0945 


1110 1275 1439 
2754 2918 | 30S2 


165 


4 


421604 1763 


2097 


2261 


2426 


2590 


164 


5 


3246 3410 


&574 


3737 


3901 


4065 


4228 


4392 4555 j 4718 


164 





4882 5045 


5203 


5371 


5534 


5697 


5860 


6023 6186 6319 


163 


7 


6511 6674 


G836 


6999 


7161 


7324 


7486 


7643 7811 7973 


162 


8 


81:35 8297 


8459 


8621 


8783 


8944 


9106 


92u3 9429 9591 


162 


9 


9752 , 9914 
43 | 
















0075 


0236 


0398 


0559 


0720 


0831 1042 | 1203 


161 



Proportional Parts. 



Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


173 


17.8 


35.6 


53.4 


71.2 


89.0 


106.8 


124.6 


142.4 


160.2 


177 


17.7 


35.4 


53.1 


70.8 


88.5 


106.2 


123.9 


141.6 


159.3 


176 


17.6 


36.2 


52.8 


70.4 


88.0 


105.6 


123.2 


140.8 


153.4 


175 


17.5 


35.0 


52.5 


70.0 


* .0 


105.0 


122.5 


140.0 


157.5 


174 


17.4 


34.8 


52.2 


69.6 


87.0 


104.4 


121.8 


139.2 


156.6 


173 


17.3 


34.6 


51.9 


69.2 


86.5 


103.8 


121.1 


138.4 


155.7 


172 


17.2 


34.4 


51.6 


68.8 


86.0 


103.2 


120.4 


137.6 


154.8 


171 


17 1 


34.2 


51.3 


68.4 


85.5 


102.6 


119.7 


136.8 


153.9 


170 


17.0 


34.0 


51.0 


68.0 


85.0 


102.0 


119.0 


136.0 


153.0 


169 


16.9 


33.8 


50.7 


67.6 


84.5 


101.4 


118.3 


135.2 


152.1 


163 


16.8 


33.6 


50.4 


67.2 


84.0 


100.8 


117.6 


1:34.4 


151.2 


167 


16.7 


33.4 


50.1 


66.8 


83.5 


100.2 


116.9 


133.6 


150.3 


166 


16.6 


&3.2 


49.8 


66.4 


83.0 


99.6 


116.2 


132.8 


149.4 


165 


16.5 


a3.o 


49.5 


66.0 


82.5 


99.0 


115.5 


132.0 


148.5 


164 


16.4 


32.8 


49.2 


65.6 


82.0 


98.4 


114.8 


131.2 


147.6 


163 


16.3 


32.6 


48.9 


65.2 


81.5 


97.8 


114.1 


130.4 


146.7 


162 


16.2 


32.4 


48.5 


64.8 


81.0 


97.2 


113.4 


129.6 


145.8 


161 


16.1 


32.2 


48.3 


64.4 


80.5 


96.6 


112.7 


128.8 


144-9 



147 



LOGARITHMS OF NUMBERS. 



No. 270 L. 431.] 



[No. 299 L. 476. 



N. 



270 
1 
2 
3 
4 
5 

6 
7 
8 
9 

280 

1 

2 
3 

4 
5 
6 

7 



431364 
2969 
4569 
6163 
7751 
9333 

440909 
2480 
4045 
5604 

7158 



450249 
1786 
3318 
4845 
6366 
7882 
9392 



460898 

2398 
3893 
5383 
6868 
8347 



471292 
2756 
4216 
5671 



1525 
3130 
4729 
6322 
7909 
9491 



1066 
2637 
4201 
5760 

7313 

8861 



1685 
3290 
4888 
6481 
8067 
9648 



1846 
3450 

5048 
6640 
8226 



1224 
2793 
4357 
5915 

7468 
9015 



0403 
1940 
3471 
4997 
6518 
8033 
9543 



1048 

2548 
4042 
5532 
7016 
8495 
9969 



1438 
2903 
4362 
5816 



0557 
2093 
3624 
5150 
6670 
8184 
9694 



1198 

2697 
4191 
5680 
7164 
8643 



0116 

1585 
3049 
4508 
5962 



1381 

2950 
4513 
6071 

7623 

9170 



2007 
3610 
5207 
6799 
8384 
9964 



0711 
2247 
3777 
5302 
6821 
8336 
9845 



1348 

2847 
4340 
5S29 
7312 
8790 



0263 
1732 
3195 
4653 
6107 



1538 
3106 
4669 
6226 

7778 
9324 



2167 
3770 
5367 
6957 
8542 



2328 
3930 
5526 
7116 

8701 



0865 
2400 
3930 
5454 
6973 
8487 
9995 



1499 

2997 
4490 
5977 
7460 



0122 

1695 
3263 
4825 
6382 

7933 

9478 



1018 
2553 
4082 
5606 
7125 
8638 



0146 
1649 

3146 

4639 
6126 
7608 
9085 



0279 
1852 
3419 
4981 
6537 



2488 
4090 
5685 
7275 

8859 



1172 

2706 
4235 

5758 
7276 

8789 



0437 
2009 
3576 
5137 
6692 

8242 
9787 



1326 
2859 
4387 
5910 
7428 
8940 



0296 
1799 

3296 
4788 
6274 
7756 
9233 



0410 


0557 


1878 


2025 


3341 


&487 


4799 


4944 


6252 


6397 



0704 
2171 
3633 
5090 
6542 



0447 
1948 

3445 

4936 
6423 
7904 
9380 



0851 
2318 
3779 
5235 



2649 
4249 
5844 
7433 
9017 



0594 
2166 
3732 
5293 
6848 

8397 
9941 



1479 
3012 
4540 
6062 
7579 
9091 



2809 
4409 
6004 
7592 
9175 



0752 
2323 
3889 
5449 
7003 

8552 

0095 
1633 
3165 
4692 
6214 
7731 
9242 



0597 
2098 

3594 

5085 
6571 
8052 
9527 



0998 
2464 
3925 
5381 
6832 



0748 
2248 

3744 
5234 
6719 
8200 
9675 



1145 
2610 
4071 
5526 
6976 



Biff. 



161 

160 
159 
159 
158 

158 
157 

157 
156 
155 

155 

154 
154 
153 
153 
152 
152 
151 

151 
150 

150 
149 
149 
148 

148 

147 
146 
146 
146 
145 



Proportional Parts. 



DilT. 


1 


§ 


3 


4 


5 


6 


7 


8 


9 


161 


16.1 


32.2 


48.3 


64.4 


80.5 


96.6 


112.7 


128.8- 


144.9 


100 


16.0 


32.0 


48.0 


64.0 


80.0 


96.0 


112.0 


128.0 


144.0 


159 


15.9 


31.8 


47.7 


63.6 


79.5 


95.4 


111.3 


127.2 


143.1 


158 


15.8 


31.6 


47.4 


63.2 


79.0 


94.8 


110.6 


126.4 


142.2 


157 


15.7 


31.4 


47.1 


62.8 


78.5 


94.2 


109.9 


125.6 


141.3 


156 


15.6 


31.2 


46.8 


62.4 


78.0 


93.6 


109.2 


124.8 


140.4 


155 


15.5 


31.0 


46.5 


62.0 


77.5 


93.0 


108.5 


124.0 


139.5 


154 


15.4 


30.8 


46.2 


61.6 


77.0 


92.4 


107.8 


123.2 


138.6 


153 


15.3 


30.6 


45.9 


61.2 


76.5 


91.8 


107.1 


122.4 


137.7 


152 


15.2 


30.4 


45.6 


60.8 


76.0 


91.2 


106.4 


121.6 


136.8 


151 


15.1 


30.2 


45.3 


60.4 


75.5 


90.6 


105.7 


120.8 


135.9 


150 


15.0 


30.0 


45.0 


60.0 


75.0 


90.0 


105.0 


120.0 


135.0 


149 


14.9 


29.8 


44.7 


59.6 


74.5 


89.4 


104.3 


119.2 


134.1 


148 


14.8 


29.6 


44.4 


59.2 


74.0 


88.8 


103.6 


118.4 


133:2 


147 


14.7 


29.4 


44.1 


58.8 


73.5 


88.2 


102.9 


117.6 


132.3 


146 


14.6 


29.2 


43.8 


58.4 


73.0 


87.6 


102.2 


116.8 


131.4 


145 


14.5 


29.0 


43.5 


58.0 


72.5 


87.0 


101.5 


116.0 


130.5 


144 


14.4 


28.8 


43.2 


57.6 


72.0 


86.4 


100.8 


115.2 


129.6 


143 


14.3 


28.6 


42.9 


57.2 


71.5 


85.8 


100.1 


114.4 


128.7 


142 


14.2 


28.4 


' 42.6 


56.8 


71.0 


85.2 


99.4 


113.6 


127.8 


141 


14.1 


28.2 


42.3 


56.4 


70.5 


84.6 


98.7 


112.8 


126.9 


140 


14.0 


28.0 


42.0 


56.0 


70.0 


84.0 


98.0 


112.0 


128.0 



148 



LOGARITHMS OF NUMBERS. 



No. 300 L. 477.1 



[No. 339 L. 531. 



300 477121 
1 8566 



310 
1 
2 
3 
4 
5 



7 
8 
9 

320 
1 
2 
3 

4 
5 
6 
7 
8 
9 

330 
1 



480007 
1443 
2874 
4300 
5721 
7138 
8551 
9958 



7266 
8711 



0151 

1586 
3016 
4442 
5863 
7280 



491362 
2760 
4155 
5544 
6930 
8311 
96S7 



0099 

1502 
2900 
4294 
5683 
7068 
8448 
9824 



501059 
2427 
3791 

5150 
6505 
7856 
9203 



1196 

2564 
3927 

5286 
6640 
7991 
9337 



510545 
1883 
3218 
4548 
5874 
7196 

8514 



521138 
2444 
3746 
5045 
6339 
7630 
8917 



0679 
2017 
3351 
4681 
6006 
7328 

8646 
9959 



1269 
2575 
3876 
5174 
6469 
7759 
9045 



7411 7555 
8855 8999 



0294 
1729 
3159 
4585 
6005 
7421 
8833 



0239 

1642 

3040 
4433 

5822 
7206 
8586 



1333 

2700 
4063 

5421 

6776 
8126 
9471 



0813 
2151 
3484 
4813 
6139 
7460 



0090 
1400 
2705 
4006 
5304 
6598 
7888 
9174 



9 530200 0328 I 0456 



0438 
1872 
3302 
4727 
6147 
7563 
8974 



0380 

1782 
3179 
4572 
5960 
7344 
8724 

0099 
1470 
2837 
4199 

5557 
6911 



0947 
2284 
3617 
4946 
6271 
7592 

8909 



7700 
9143 



0582 



7844 
9287 



0725 



9431 



9 ! Diff. 



8133 8278 8422 
9575 | 9719 9863 



2016 2159 2302 2445 
3445 3587 ' 3730 



1012 1156 
2588 



4869 
6289 
7704 
9114 



5011 
6430 
7845 
9255 



5153 5295 

6572 6714 

986 8127 



4015 
5437 
6855 
8269 



9396 9537 £67? 



0520 0661 i 0801 0941 



1922 
3319 
4711 
6099 

7483 
8862 

0236 
1607 
2973 
4335 

5693 
7046 
8395 
9740 

1081 
2418 
3750 
5079 
6403 
7724 

9040 



0374 
1744 
3109 
4471 

5828 
7181 
8530 
9874 



1081 

2481 

S876 

5267 

6515 1 6653 

7759 7897 8035 

9137 9275 9412 



2062 2201 2341 

3458 3597 3737 

4850 4989 5128 

6238 

7621 

8999 



0511 0648 | 0785 

1880 2017 2154 

3246 I 3382 £518 

4607 4743 | 4878 



5964 
T316 



6099 : 6234 
7451 | 7586 



8664 8799 8934 



0221 
1530 
2835 
4136 
54:34 
6727 
8016 
9302 



0.353 
1661 
2966 
4266 
5563 
6856 
8145 
9430 



1215 
2551 

3883 
5211 
65&5 
7855 

9171 



0009 0143 

1349 1482 

2684 2818 

4016 : 4149 

5344 5476 

6668 I 68C0 

7987 | 8119 

9303 9434 



C277 
1616 
2951 
4282 
5CC9 
6932 
8251 

£566 



1299 
2731 
4157 
5579 
6997 
8410 



1222 

2621 
4015 
5406 
6791 
8173 
9550 



0922 
2291 
£655 
5014 

6370 
7721 

CC68 



0484 


0615 


1792 


1922 


3096 


3226 


4396 


4526 


5693 


5822 


6985 


7114 


8274 


8402 


9559 


9687 



0745 
2053 



0876 

2183 
3356 I 3486 
4656 4785 
5951 6C81 
7243 , 7372 

8531 eeco 
9815 ! 9943 



0584 I 0712 |l 0840 ! 0968 [ 1096 1223 



0411 
1750 

£C84 
4415 
5741 
7C64 

8£82 

CC97 

1CC7 
2314 
£616 
4915 
C210 
7501 
8788 



0072 
1351 



145 
144 

144 
143 
143 
142 
142 
141 
141 

140 

140 

139 
139 
139 
138 
138 

137 
137 
136 
136 

136 
135 
135 

134 
134 
183 

133 
133 

1S2 
1S2 

131 

131 
131 

130 
1£0 
129 
129 
129 

128 

128 



Proportional Parts. 



Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


139 


13.9 


27.8 


41.7 


55.6 


69.5 


83.4 


97.3 


111.2 


125.1 


138 


13.8 


27.6 


41.4 


55.2 


69.0 


82.8 


96.6 


110.4 


124.2 


137 


13.7 


27.4 


41.1 


54.8 


68.5 


82.2 


95.9 


109.6 


123.3 


136 


13.6 


27.2 


40.8 


54.4 


68.0 


81.6 


95.2 


108.8 


122.4 


135 


13.5 


27.0 


40.5 


54.0 


67.5 


81.0 


94.5 


108.0 


121.5 


134 


13.4 


26.8 


40.2 


53.6 


67.0 


80.4 


93.8 


107.2 


120.6 


133 


13.3 


26.6 


39.9 


53.2 


66.5 


79.8 


93.1 


106.4 


119.7 


132 


13.2 


26.4 


39.6 


52.8 


66.0 


79.2 


92.4 


105.6 


118.8 


131 


13.1 


26.2 


39.3 


52.4 


65.5 


78.6 


91.7 


104.8 


117.9 


130 


13.0 


26.0 


39.0 


52.0 


65.0 


78.0 


91.0 


104.0 


117.0 


129 


12.9 


25.8 


38.7 


51.6 


64.5 


77.4 


90.3 


103.2 


116.1 


128 


12.8 


25.6 


as. 4 


51.2 


64.0 


76.8 


89.6 


102.4 


115.2 


127 


12 7 


25.4 


38.1 


50.8 


63.5 


76.2 


88.9 


101.6 


114.3 



149 



LOGARITHMS OF NUMBERS. 



No. 340 L. 531.] 



[No. 379 L. 579. 



N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 


340 


531479 


1607 


1734 


1862 


1990 


2117 2245 2372 


2500 


2627 


128 


1 


2754 


2882 


3009 


3136 


3264 


3391 ' 3518 3645 


3772 


3899 


127 


2 


4026 


4153 


4280 


4407 


4534 


4661 i 4787 


4914 


5041 


5167 


127 


3 


5294 


5121 


5547 


5674 


5800 


5927 i 6053 


6180 


6306 


6432 


126 


4 


6558 


6685 


6811 


6937 


7063 


: 7189 7315 


7441 


7567 


7693 


126 


5 


7619 


7945 


8071 


8197 


8322 


8448 | 8574 


8699 


8825 


8951 


126 




9076 


9202 


9327 


9452 


9578 


9r03 ; 9829 


9954 








6 
























0079 
1330 


0204 
1454 


125 
125 


7 


540329 


0155 


0580 


0705 


0830 


0955 . 1080 


1205 


8 


1579 


1704 


1829 


1953 


2078 


2203 1 2327 


2452 


2576 


2701 


125 


9 


2825 


2950 


3074 


3199 


3323 


3447 


3571 


3696 


3820 


3944 


124 


350 


4068 


4192 


4316 


4440 


4564 


: . 4688 


4812 


4936 


5060 


5183 


124 


1 


5307 


5431 


5555 


5678 


5802 


, 5925 


6049 


6172 


6296 


6419 


124 


2 


6543 


6666 


6789 


6913 


7036 


! 7159 


7282 


7405 


7529 


7652 


123 


3 


7775 


7898 


8021 


8144 


8267 


8389 


8512 


8635 


8758 


8881 


123 


4 


9003 


9126 


9249 


9371 


9494 


9616 


9739 


9861 


9984 














0106 
1328 


123 
122 


5 


550228 


0351 


0473 


0595 


0717 


0840 


0962 


1084 


1206 


6 


1450 


1572 


1694 


1816 


1938 


; 2060 


2181 


2303 


2425 


2547 


122 




2668 


2790 


2911 


3033 


3155 


1 3276 


3398 


3519 


3640 


3762 


121 


8 


3883 


4004 


4126 


4247 


4368 


4489 


4610 


4731 


4852 


4973 


121 


9 


5094 


5215 


5336 


5457 


5578 


5699 


6820 


5940 


6061 


6182 


121 


360 


6303 


6423 


6544 


6664 


6785 


6905 


7026 


7146 


7267 


7387 


120 


1 


7507 


7627 


7748 


7868 


7988 


8108 


8228 


sai9 


8469 


&589 


120 


2 
3 


8709 
9907 


8829 


8948 


9068 


9188 


9308 


9428 


9548 


9667 


9787 


120 


0026 


0146 


0265 


0385 


0504 


0624 


0743 


0863 


0982 
2174 


119 


4 


561101 


1221 


1340 


1459 


1578 


1698 


1817 


1936 


2055 


119 


5 


2293 


2412 


2531 


2650 


2769 


2887 


3006 


3125 


3244 


3362 


119 


6 


3481 


3600 


3718 


3837 


3955 


1 4074 


4192 


4311 


4429 


4548 


119 


7 


4666 


4784 


4903 


5021 


5139 


' 5257 


5376 


5494 


5612 


5730 


118 


8 


5848 


5966 


6084 


6202 


6320 


! 6437 


6555 


6673 


6791 


6909 


118 


9 


7026 


7144 


7262 


7379 


7497 


7014 


7732 


7849 


7967 


8084 


118 


370 


8202 


8319 


8436 


8554 


8671 


8788 


8905 


9023 


9140 


9257 


117 


1 


9374 


9491 


9fi0fi 


Q'VOK 


9842 


9959 










t/UV/O %j i vl; 


0076 


0193 


0309 0426 
1476 | 1592 


117 
117 


2 


570543 


0660 


0776 


0893 


1010 


, 1126 


1243 


1359 


3 


1709 


1825 


1942 


2058 


2174 


i 2291 


2407 2523 


2639 2755 


116 


4 


2872 


2988 


3104 


3220 


3336 


3452 


3568 3684 


3800 1 3915 


116 


5 


4031 


4147 


4263 


4379 


4494 


i 4610 


4726 ! 4841 


4957 


5072 


116 


• 6 


5188 


5303 


5419 


5534 


5650 


' 5765 


5880 i 5996 


6111 


6226 


115 


7 


6341 


6457 


6572 


6687 


6802 


I 6917 


7032 7147 


7262 


7377 


115 


8 


7492 


7607 


7722 


7836 


7951 


8066 


8181 


8295 


&410 


8525 


115 


9 


8639 


8754 


8868 


8983 


9097 


9212 

I 


9326 


9441 


9555 


9669 


114 



Proportional Parts. 



Diff. 


1 


2 


3 


< 


5 


6 


7 


8 


128 


12.8 


25.6 


38.4 


51.2 


64.0 


76.8 


89.6 


102.4 


127 


12.7 


25.4 


38.1 


50.8 


63.5 


76.2 


88.9 


101.6 


126 


12.6 


25.2 


37.8 


50.4 


63.0 


75.6 


88.2 


100.8 


125 


12.5 


25.0 


37.5 


50.0 


62.5 


75.0 


87.5 


100.0 


124 


12.4 


24.8 


37.2 


49.6 


62.0 


74.4 


86.8 


99.2 


123 


12.3 


24.6 


36.9 


49.2 


61.5 


73.8 


86.1 


98.4 


122 


12.2 


24.4 


36.6 


48.8 


61.0 


73.2 


85.4 


97.6 


121 


12.1 


24.2 


36.3 


48.4 


60,5 


72.6 


84.7 


96.8 


120 


12.0 


24.0 


36.0 


48.0 


60.0 


72.0 


84.0 


96.0 


119 


11.9 


23.8 


35.7 


47.6 


59.5 


71.4 


83.3 


95.2 | 



150 



LOGARITHMS OF NUMBERS. 



No. 380. L. 579.] 



[No. 414 L. 617. 



N. 





'-I 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 




579784 


1 


! ! 1 


1 




1 






380 


9898 ' 


0012 | 0126 0241 
1153 1207 ! 1381 i 


0355 0469 0583 


0697 


0811 
1950 


114 


1 


580925 


1039 


1495 ' 1608 1722 ( 1836 




f, 


2063 


2177 


2291 2404 2518 


21531 2745 2858 


2972 


3085 




3 


3199 


3312 


3426 3539 


3652 


3765 ! 3879 3992 


4105 


4218 




4 


4331 


4444 


4557 4670 


4783 | 


4896 5009 5122 


5235 


5348 


113 


5 


5461 


5574 


56<6 


5799 


5912 ; 


6004 6137 6250 


6362 


6475 




6 


6587 


6700 


6812 


6925 


7037 


7149 , 7262 7374 7486 


7599 




7 


7711 


7823 


7935 


8047 


8160 


8272 8384 8490 


8720 


112 


8 


8832 


8944 


9056 9167 


9279 


9391 9503 9615 | 9726 


9838 




9 


9950 












0061 


0173 | 0284 


0396 


0507 0619 0730 0842 


0953 




390 


591065 


1176 


1287 ' 1399 


1510 


1621 1732 1843 ' 1955 


2066 




1 


2177 


2288 


2399 ; 2510 


2621 


2732 2843 2954 3064 


3175 


111 


a 


3286 


3397 


3508 3618 


3729 


3840 39.50 4061 4171 


4282 




3 


4393 


4503 


4614 4724 


4834 


4945 ( 5055 5165 5276 


5386 




4 


5496 


5606 5717 


5937 


6047 6157 6267 | 6377 


6487 


110 


5 


6597 


6707 


6817 i 6927 


7037 


7146 


7256 7366 ! 7476 


7586 


6 


7695 


7805 


7914 8024 


8134 


8043 


8353 8462 8572 


8681 




7 


8791 


8900 


9009 i 9119 


9228 


9:337 


9446 9556 9665 


9774 




8 


9688 


9992 












109 


0101 


0210 


C319 


0428 0537 0646 1 0755 


086* 


9 


600973 


1082 


1191 


1299 


1408 


1517 i 1625 | 1734 | 1843 


1951 




400 


2060 


2169 


2277 


2386 


2494 


2603 1 2711 2819 


3036 




1 


3144 


3253 


3331 


3469 


3577 


3794 3902 4010 


4118 


108 


2 


4226 


4334 


4442 


4550 


4658 


4766 4874 49>- 


5197 




3 




5413 


5521 


5736 


5844 5951 6059 i 6166 


6274 




4 


6381 


6489 


6596 J 6704 


6811 


6919 7026 7133 7241 


7348 




5 


7455 


7562 


7669 :::: 


7884 


7991 8098 8205 8312 


8419 


107 


6 


B526 


8633 


8740 8847 


8954 


9061 9167 9274 9381 


9488 




7 


9594 


9701 


9808 9914 










0021 
1086 


0128 0234 0341 0447 
1192 1298 1405 1511 


0554 




8 


610660 


0707 


0873 


0979 


1617 




9 


1723 


1829 


1936 


2042 


2148 


22.54 2360 2466 2572 


2678 


106 


410 


27-4 


2890 


2996 


3102 


3207 


3313 3419 3525 3630 


3736 




1 


3842 


3947 


4053 4159 


4264 


4370 4475 4581 4686 






2 


4897 


5003 


5213 


5319 


5424 5529 5634 5740 


5845 




3 


5950 


6055 


6160 6265 


6370 


6476 6581 6686 6790 


6895 


105 


4 


7000 


7105 


7210 7315 


7420 


7525 7629 77:34 7839 


7943 





Proportional. Parts. 



Diff. 


1 


o 


3 


4 


5 


6 


7 


8 


9 


118 


11.8 


23.6 


£5.4 


47.2 


59.0 


70.8 


82.Q 


94.4 


106.2 


117 


11.7 


23.4 


35.1 


46.8 


58.5 


70.2 


81.9 


93.6 


105.3 


116 


11.6 


23.2 


34.8 


46.4 


58.0 


69.6 


81.2 


92.8 


104.4 


115 


11.5 


23.0 


34.5 


46.0 


57.5 


69.0 


80.5 


92.0 


103.5 


114 


11.4 


22.8 


34.2 


45.6 


57.0 


68.4 


79.8 


91.2 


102. Q 


113 


11.3 


22.6 


33.9 


45.2 


56.5 


67.8 


79.1 


90.4 


101.7 


112 


11.2 


22.4 


33.6 


44.8 


56.0 


67.2 


78.4 


89.6 


100.8 


111 


11.1 


22.2 


33.3 


44.4 


55.5 


66.6 


77.7 


88.8 


99.9 


110 


11.0 


22.0 


33.0 


44.0 


55.0 


66.0 


77.0 


88.0 


99.0 


109 


10.9 


21.8 


32.7 


43.6 


54.5 


65.4 


76.3 


87.2 


98.1 


108 


10.8 


21.6 


32.4 


43.2 


54.0 


64.8 


75.6 


86.4 


97.2 


107 


10.7 


21.4 


32.1 


42.8 


53.5 


64.2 


74.9 


85.6 


96.3 




10.6 


21.2 


31.8 


42.4 


53.0 


63.6 


74.2 


84.8 


95.4 




10.5 


21.0 


31.5 


42.0 


52.5 


63.0 


73.5 


84.0 


94.5 


105 


10.5 


21.0 


31.5 


4AJ.0 


5<2.5 


63.0 


73.5 


84.0 


94.5 


104 


10.4 


20.8 


31.2 


41.6 


52.0 


62.4 


72.8 


83.2 


93.6 



151 



LOGARITHMS OF NUMBERS. 



No. 415 L. 618.] 



[No. 459 L. 662 



N. 



415 
6 



9 

420 
1 
2 
3 
4 
5 



7 
8 
9 

450 
1 
2 
3 
4 
5 



618048 
9093 



620136 
1176 
2214 

3249 

4282 
5312 
6340 
7366 
8389 
9410 



630428 
1444 

2457 

34G8 
4477 
5484 
6488 
7490 
8489 
9486 



640481 
1474 
2465 

3453 

4439 
5422 
6404 
7383 
8360 



650308 
1278 
2216 

3213 

4177 
5138 
6098 
7056 
8011 
8965 
9916 



660865 
1813 



8153 ' 8257 
9198 i 9302 



0240 0344 
1280 I 1384 
2421 



2318 

3353 

4385 
5415 
6443 
7468 
8491 
9512 



3456 
4488 
5518 
6546 
7571 
8593 
9613 



0530 
1545 
2559 

3569 
4578 
55&4 
6588 
7590 
8589 



0631 
1647 
2660 

3670 
4679 
5685 
6688 
7690 
8689 



0581 
1573 
2563 

3551 
4537 
5521 

6502 
7481 
8458 
9432 

0405 
1375 
2343 

3309 
4273 
5235 
6194 
7152 
8107 
9060 



0680 
1672 
2662 

3650 
4636 
5619 
6600 
7579 
8555 
9530 



0011 
0960 
1907 



0502 
1472 

2440 

3405 
4369 
5331 
6290 
7247 
8202 
9155 



0106 
1055 
2002 



8362 
9406 



9511 



0733 

1748 
2761 

3771 
4779 

5785 
6789 
7790 
8789 
9785 



0779 
1771 
2761 

3749 
4734 
5717 
6698 
7676 
8653 
9627 



0835 
1849 
2862 

3872 
4880 
5886 
6889 
7890 
8888 
9885 



0879 
1871 
2860 

3847 
4882 
5815 
6796 
7774 
8750 
9724 



0599 
1569 
2536 

3502 
4465 
5427 
6386 
7343 
8298 
9250 



0201 
1150 
2096 



0696 
1666 
2633 

3598 

4562 
5523 
6482 
7438 
8393 
9346 



0296 
1245 
2191 



8571 
9615 



0448 


0552 


1488 


1592 


2525 


2628 


3559 


3663 


4591 


4695 i 


5621 


5724 


6648 


6751 


7673 


7775 


8695 


8797 


9715 


9817 



0656 
1695 
2732 

3766 
47\J8 
5827 
6853 
7878 
8900 
9919 

0936 
1951 
2963 

3973 
4981 
5986 
C989 
7990 
8988 
9984 



9719 

0760 
1799 
2835 

3869 
4901 
5929 
6956 

7980 
9002 



8780 8684 
9824 9928 



0021 

1038 
2052 
3064 

4074 

5081 
6087 
7C89 
8090 
9088 



CC78 
1970 
2959 

3946 
4931 
5913 
6894 

7872 
8848 
9821 



0793 
1762 
2730 

3695 
4658 
5619 
6577 
7534 
8488 
9441 



0084 
1077 
2069 
3058 

4044 
5029 
6011 
6992 
7969 
6945 
9919 



1859 
2826 

3791 

4754 
5715 
667'3 
7629 
8584 
9536 



0391 i 0486 
1339 ' 1434 
2286 I 2380 



0864 


0968 


1903 


2007 


2939 


3042 


3973 


4076 


5004 


5107 


6032 


6135 


7058 


7161 


8082 


8185 


9104 


9206 



0183 
1177 
2168 
3156 

4143 
5127 
6110 

7089 
8067 
9043 



0016 
0987 
1956 
2923 

3888 
4850 
5810 
6769 
7725 
8679 
9631 



0581 
1529 
2475 



0283 
1276 
2267 
3255 

4242 
5226 
6208 
7187 
8165 
9140 



0113 

1084 
2053 
8019 

3984 
4946 
5906 
6864 
7820 
8774 
9726 



0676 
1623 
2569 



8989 

0032 
1072 
2110 
3146 

4179 

5210 
6238 

7263 
8287 



0123 


0224 


0326 


1139 


1241 


1342 


2153 


2255 


2356 


3165 


3266 


3367 


4175 


4276 


4376 


5182 


5283 


5883 


6187 


6287 


6388 


7189 


7290 


7890 


8190 


82S0 


8889 


9188 


9287 


9387 



0382 
1375 
2366 
3354 

4340 

5324 
6306 
7'285 
8262 
8237 



0210 
1181 
2150 
3116 

4080 
5042 
6002 
6960 
7916 
8870 
9821 



0771 
1718 
2663 



Proportional, Parts. 



Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


105 


10.5 


21.0 


31.5 


42.0 


52.5 


63.0 


73.5 


84.0 


94.5 


104 


10.4 


20.8 


31.2 


41.6 


52.0 


62.4 


72 8 


83.2 


93.6 


103 


10.3 


20.6 


30.9 


41.2 


51.5 


61.8 


72 1 


82.4 


92.7 


102 


10.2 


20.4 


30.6 


40.8 


51.0 


61.2 


71 4 


81.6 


91.8 


101 


10.1 


20.2 


30.3 


40.4 


50.5 


60.6 


70 7 


80.8 


90.9 


100 


10.0 


20.0 


30.0 


40.0 


50.0 


60.0 


70 


80.0 


90.0 


99 


9.9 


19.8 


29.7 


39.6 


49.5 


59.4 


69.3 


79.2 


89.1 



152 



LOGARITHMS OF NUMBERS. 



No.' 460 L. 662.] 



[No. 499 L. < 



N 



460 

1 
2 
3 

4 
5 



470 
1 



9 

480 
1 
2 
3 
4 
5 
6 
7 
8 
9 

490 
1 
2 
3 
4 
5 
6 
7 
8 
9 



662758 
3701 
4642 
5581 
6518 
7453 
8386 
9317 



2852 
3795 
4736 
5675 
6612 
7546 
8479 
9410 



2947 
3889 
4830 
5769 
6705 
7640 
8572 
9503 



3041 
3983 
4924 

5862 
6799 



3135 

4078 
5018 
5956 
6892 



1733 ! 7826 
8665 i 8759 
9596 9689 



670246 I 0339 
1173 , 1265 

2098 I 2190 

3021 3113 

3942 4034 

4861 4953 



5778 
6694 
7607 
8518 
9428 



5870 
6785 
7698 
8609 
9519 



0431 ! 0524 

1358 1451 

2283 ! 2375 

3205 3297 

4126 4218 

5045 5137 

5962 6053 

6876 | 6968 

7789 7881 

8700 8791 

9610 | 9700 



0617 
1543 

2467 
3390 
4310 

5228 
6145 

7059 
7972 
8882 
9791 



680336 

1241 

2145 
3047 
3947 
4845 
5742 
6636 
7529 
8420 
9309 



690196 
1081 
1965 
2847 
3727 
4605 
5482 
6356 
7229 
8100 



0426 

1332 

2235 
3137 
4037 
4935 
5831 
6726 
7618 
8509 
9398 



0285 
1170 
2053 
2935 
3815 
4693 
5569 
6444 
7317 
8188 



0517 

1422 

2326 
3227 
4127 
5025 
5921 
6815 
7707 
8598 
9486 



0607 

1513 

2416 
3317 
4217 
5114 
6010 
6904 
7798 
8687 
9575 



0373 
1258 
2142 
3023 
3903 
47'81 
5657 
6531 
7404 
8275 



0462 
1347 
2230 
3111 
3991 
4868 
5744 
6618 
7491 
8362 



5 



3230 

4172 
5112 
6050 
6986 
7920 
8852 
9782 



0710 
1636 

2560 
3482 
4402 
5320 
6236 
7151 
8063 
8973 
9882 



0698 

1603 
2506 
3407 
4307 
5204 
6100 
6994 
7'8S6 
8776 
9664 



0789 

1693 
2596 
3497 
4396 
5294 
6189 
7083 
7975 
8865 
9753 



0550 
1435 
2318 
3199 
4078 
4956 
5832 
6706 
7578 
8449 



0639 
1524 
2406 
3287 
4166 
5044 
5919 
6793 
7665 
8535 



6 



&324 
4266 
5206 
6143 
7079 
8013 
8945 
9875 



0802 
1728 

2652 
3574 
4494 
5412 
6328 
7242 
8154 
9064 
9973 



3418 
4360 
5299 
6237 
7173 
8106 
9038 
9967 



3512 
4454 
5393 
6331 
7266 
8199 
9131 



0379 

1784 
2686 
35S7 
4486 
5:383 
6279 
7172 
8064 
8953 
9841 



0895 
1821 

2744 
3666 
4586 
5503 
6419 
7333 
8245 
9155 



0728 
1612 
2494 
3375 
4254 
5131 
6007 
6880 
7752 



0063 
0970 

1874 
2777 
3677 
4576 
5473 
6368 
7261 
8153 
9042 
9930 



0316 
1700 
2583 
3463 
4342 
5219 
6094 
6968 
7839 
8709 



0060 
0988 
1913 

2836 
3758 
4677 
5595 
6511 
7424 
8336 
9246 



0154 
1060 

1964 
2867 
3767 
4666 
5563 
6458 
7351 
8242 
9131 



0019 

0905 
1789 
2671 
8551 
4430 
5307 
6182 
7055 
7926 
8796 



3607 
4548 
5487 
6424 
7360 
8293 
9224 



0153 
1080 
2005 

2929 
3S50 
4769 
5687 
6602 
7516 
8427 
9337 



0245 
1151 

2055 
2957 
3657 
4756 
5652 
6547 
7440 
8331 
9220 



0107 

0993 
1877 
2759 
3639 
4517 
5394 
6269 
7142 
8014 
8833 



Diff. 



94 



93 



91 



90 



89 



87 



Proportional Parts. 



Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


98 


9.8 


19.6 


29.4 


39.2 


49.0 


58.8 


68.6 


78.4 


88.2 


97 


9.7 


19.4 


29.1 


38.8 


48.5 


58.2 


67.9 


77.6 


87.3 


96 


9.6 


19.2 


28.8 


38.4 


48.0 


57.6 


67.2 


76.8 


86.4 


95 


9.5 


19.0 


23.5 


33.0 


47.5 


57.0 


66.5 


76.0 


85.5 


94 


9.4 


18.8 


23.2 


37.6 


47.0 


56.4 


65.8 


75.2 


84.6 


93 


9.3 


18.6 


27.9 


37.2 


46.5 


55.8 


65.1 


74.4 


83.7 


92 


9.2 


18.4 


27.6 


36.8 


46.0 


55.2 


64.4 


73.6 


82.8 


91 


9.1 


18.2 


27.3 


36.4 


45.5 


54.6 


63.7 


72.8 


81.9 


90 


9.0 


18.0 


27.0 


36.0 


45.0 


54.0 


63.0 


72.0 


81.0 


89 


8.9 


17.8 


26.7 


35.6 


44.5 


53.4 


62.3 


71.2 


80.1 


88 


8.8 


17.6 


26.4 


35.2 


44.0 


52.8 


61.6 


70.4 


79.2 


87 


8.7 


17.4 


26.1 


34.8 


43.5 


52.2 


"60.9 


69.6 


78.3 


86 


8.6 


17.2 


25.8 


34.4 


43.0 


51.6 


60.2 


68.8 


77.4 



153 



LOGARITHMS OF NUMBERS. 



No. 500 L. 698.] 








[N 


o. 544 L. 736. 


N. 





1 


2 


3 


4 1 


5 


6 


7 


8 


9 


Diff. 


500 


698970 


9057 


9144 


9231 


9317 ' 


9404 


9491 


9578 


9664 


9751 




1 


. 9838 


9924 




















0011 
0877 


0098 
0963 


0184 i 
1050 ! 


0271 
1136 


0358 
1222 


0444 
1309 


0531 
1395 


0617 
1482 




2 


700704 


0790 




3 


1568 


1654 


1741 


1827 


1913 


1999 


2086 


2172 


2258 


2344 




4 


2431 


2517 


2603 


2689 


2775 


2861 


2947 


3033 


3119 


3205 




5 


3291 


3377 


3463 


3549 


3635 


3721 


3807 


3893 


3979 


4065 


86 


6 


4151 


4236 


4322 


4408 


4494 


4579 


4665 


4751 


4837 


4922 




7 


5008 


5094 


5179 


5265 


5350 


5436 


5522 


5607 


5693 


5778 




8 


5864 


5949 


6035 


6120 


6206 


6291 


6376 


6462 


6547 


6632 




9 


6718 


6803 


6888 


6974 


7059 


7144 


7229 


7315 


7400 


7485 




510 


7570 


7655 


7740 


7826 


7911 


7996 


8081 


8166 


8251 


8336 


85 


1 


&421 


8506 


8591 


8676 


8761 


8846 


8931 


9015 


9100 


9185 


2 


9270 


9355 


9440 


9524 


9609 


9694 


9779 


9863 


9948 












0033 
0879 




3 


710117 


0202 


02S7 


0371 


0450 


0540 


0625 


0710 


0794 




4 


0963 


1048 


1132 


1217 


1301 


1S85 


1470 


1554 


1639 


1723 




5 


1807 


1832 


1976 


2060 


2144 


2229 


2313 


2397 


2481 


2566 




6 


2050 


2734 


2818 


2902 


2986 


3070 


3154 


3238 


3323 


3407 


84 


7 


3491 


3575 


3659 


3742 


3826 


3910 


3994 


4078 


4162 


4246 


8 


4330 


4414 


4497 


4581 


4665 


4749 


4833 


4916 


5000 


5084 




9 


5167 


5251 


5335 


5418 


5502 


5586 


5669 


5753 


5836 


5920 




520 


6003 


6087 


6170 


6254 


6337 


6421 


6504 


6588 


6671 


6754 




1 


6838 


6921 


7004 


7088 


7171 


7254 


7338 


7421 


7504 


7587 




2 


7671 


7754 


7'837 


7920 


8003 


8086 


8169 


8253 


8336 


8419 


83 


3 


8502 


8585 


8668 


8751 


8834 


8917 


9000 


9083 


9165 


9248 


4 


9331 


9414 


9497 


9580 


9663 


9745 


9828 


9911 


9994 






0077 
0903 




5 


720159 


0242 


0325 


0407 


0490 


0573 


0655 


0738 


0821 




6 


0986 


1068 


1151 


1233 


1316 


1393 


1481 


1563 


1646 


1728 




7 


1811 


1893 


1975 


2058 


2140 


2222 


2305 


2387 


2469 


2552 




8 


2634 


2716 


2798 


2881 


2963 


3045 


3127 


3209 


3291 


3374 




9 


3456 


3538 


3620 


3702 


3784 


3866 


3948 


4030 


4112 


4194 


82 


530 


4276 


4358 


4440 


4522 


4604 


4665 


4767 


4849 


4931 


5013 




1 


5095 


5176 


5258 


5340 


5422 


5503 


5585 


5667 


5748 


5830 




2 


5912 


5993 


6075 


6156 


6238 ! 


6320 


6401 


6483 


6564 


6646 




3 


6727 


6809 


6890 


6972 


7053 ; 


7134 


7216 


7297 


7379 


7460 




4 


7541 


7623 


7704 


7785 


7'866 


7948 


8029 


8110 


8191 


8273 




5 


8354 


8435 


8516 


8597 


8678 


8759 


8841 


8922 


9003 


9084 




6 


9165 


9246 


9327 


9408 


9489 i 


9570 


9651 


9732 


9813 


9893 


81 


7 


9974 






















0055 
0863 


0136 
0944 


0217 
1024 


0298 


0378 
1186 


0459 
1266 


0540 
1347 


0621 


0702 




8 


730782 


1105 


1428 


1508 




9 


1589 


1069 


1750 


1830 


1911 


1991 


2072 


2152 


2233 


2313 




540 


2394 


2474 


2555 


2635 


2715 


2796 


2876 


2956 


3037 


3117 




1 


3197 


3278 


3358 


'3438 


3518 


3598 


3679 


3759 


3839 


3919 




2 


3999 


4079 


4160 


4240 


4320 


4400 


4480 


4560 


4640 


4720 


80 


3 


4800 


4880 


4960 


5040 


5120 


5233 


5279 


5359 


5439 


5519 


4 


5599 


5679 


5759 


5838 


5918 


5998 


6078 


6157 


6237 


6317 






Pro 


PORTIONAL PA 


RTS. 










Diff. 


1 


2 


3 


4 


5 


6 


' 


8 


9 


87 


8.7 


17.4 


26.1 


« 


54.8 


43.5 


52.2 




60 


.9 




69.6 


78.3 


86 


8.6 


17.2 


25.8 


I 


54.4 


43.0 


51.6 




60 


.2 




68.8 


77.4 


85 


8.5 


17.0 


25.5 


j 


J4.0 


42.5 


51.0 




59 


.5 




68.0 


76.5 


84 


8.4 


16.8 


25.2 


1 


53.6 


42.0 


50.4 




58 


.8 




67.2 


75.6 



154 



LOGARITHMS OF NUMBERS. 



No. 545 L. 736.1 



|No. 584 L. 707. 



N. 






1 


2 


8 


4 



545 
6 

7 



550 
1 



560 
1 
2 

3 

4 
5 
6 
7 
8 
9 

570 
1 
2 
3 
4 
5 

6 

7 
8 
9 

580 
1 
2 
3 

4 



736397 
7193 
7987 
8781 
9572 



740363 
1152 
1939 
2725 
3510 
4293 
5075 
5855 
6634 
7412 

8188 
8963 
9736 



6476 6556 
7272 7352 
8067 8146 
8660 8939 
9651 I 9731 



0442 ! 0521 

1230 j 1309 

2018 2096 

2804 ' 2882 

3588 3667 

4371 4449 

5153 \ 5231 

5933 6011 

6712 6790 

7489 j 7567 

8266 8343 

9040 9118 



6635 6715 
7431 ! 7511 
8225 8305 
9018 9097 
9810 9889 



0600 
1388 
2175 
2961 
3745 
4528 
5309 



7645 

8421 
9195 



0678 
1467 
2254 
3039 
3623 
4606 
5387 
6167 
6945 
7722 

8498 
9272 



750508 
1279 
2048 
2816 
3583 
4348 
5112 

5875 
6636 
7396 
8155 
8912 
9668 



760422 
1176 
1928 
2679 

3428 
4176 
4923 
5669 
6413 



0586 0663 0740 

1356 1433 j 1510 

2125 2202 2279 

2893 2970 3047 

3660 3736 ! 3813 

4425 4501 4578 

5189 5265 5341 



5951 I 6027 
6712 6788 
7472 7548 
8230 8306 
8988 9063 
9743 i 9819 



6103 
6864 
7624 
8382 
9139 



0498 I 0573 I 0649 

1251 1326 1402 

2003 2078 2153 

2754 2829 2904 

3503 3578 3653 

4251 j 4326 ' 4400 

4998 5072 5147 

5743 5818 5892 

6487 6562 6636 



0045 
0817 
1587 
2356 
3123 
3889 
4654 
5417 

6180 
6940 
7700 
8458 
9214 
9970 



0724 

1477 
2228 
297'8 

3727 
4475 
5221 
5966 
6710 



6795 
7590 



9177 



0757 
1546 
2332 
3118 
3902 
4684 
5465 
6245 
7023 
7800 

8576 
9350 



6 



6874 
7670 
8463 
9256 



0047 

0836 
1624 
2411 
3196 
3980 
4762 
5543 
6323 
7101 
7878 

8653 
9427 



0123 

C894 
1604 
2433 
3200 
3966 
4730 
5494 

6256 

7016 
7775 
8533 
9290 



0200 
0971 
1741 
2509 

S277 
4042 
4807 
5570 

6332 
7092 
7851 
8609 
9366 



6954 
7749 
8543 
9335 



0126 

0915 
1703 
2489 
3275 
4058 
4&40 
5621 
6401 
7179 
7955 

8731 
9504 



8 



I 



9 ! Diff. 



7034 ! 7113 

7829 7908 

8622 8701 

9414 9493 



0205 | 0284 

0994 | 1073 

1782 i 1860 

2568 I 2647 

3353 I 3431 

4136 | 4215 

4919 I 4997 

5699 5777 

6479 6556 

7256 7334 

8033 8110 

8808 8885 

9582 I 9659 



0277 
1048 
1818 
2586 
3353 
4119 
4683 
5646 

6408 
7168 
7927 
8685 
9441 



0.354 
1125 
1895 
2663 
84 SO 
4135 
4960 
5722 

6484 
7244 
8003 
8761 
9517 



0431 
1202 
1972 
2740 
3506 
4272 
5036 
5799 

6560 

7320 
8079 
8836 
9592 



0045 


0121 


0799 


0875 


1552 


1627 


2303 


2378 


3053 


3128 


3802 


3877 


4550 


4624 


5296 


5370 


6041 


6115 


, 6785 


6859 



0196 
0950 
1702 
2453 
3203 

3952 
4699 
5445 
6190 
6933 



0272 
1025 
1778 
2529 
3278 

4027 
4774 
5520 
6264 
7007 



0347 
1101 
1853 
2604 
3353 

4101 
4848 
5594 
6338 
7082 



Proportional Parts. 



Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


83 


8.3 


16.6 


24.9 


33.2 


41.5 


49.8 


58.1 


66.4 


74.7 


82 


8.2 


16.4 


24.6 


32.8 


41.0 


49.2 


57.4 


65.6 


73.8 


81 


8.1 


16.2 


24.3 


32.4 


40.5 


48.6 


56.7 


64.8 


72.9 


80 


8.0 


16.0 


24.0 


32.0 


40.0 


48.0 


56.0 


64.0 


72.0 


79 


7.9 


15.8 


23.7 


31.6 


39.5 


47.4 


55.3 


63.2 


71.1 


78 


7.8 


15.6 


23.4 


31.2 


39.0 


46.8 


54.6 


62.4 


70.2 


77 


7.7 


15.4 


23.1 


30.8 


38.5 


46.2 


53.9 


61.6 


69.3 


76 


7.6 


15.2 


22.8 


30.4 


38.0 


45.6 


53.2 


60.8 


68.4 


75 


7.5 


15.0 


22.5 


30.0 


37.5 


45.0 


52.5 


60.0 


67.5 


74 


7.4 


14.8 


22.2 


29.6 


37.0 


44.4 


51.8 


59.2 


[ 66.6 



155 



LOGARITHMS OF NUMBERS. 



No. 


585 L. 767.1 














[N 


o. 629 L, 799. 


N. 





1 


2 


8 


4 


5 6 


7 


8 


9 


Diff. 


585 


767156 


7230 7304 


7379 


7453 


7527 7601 


7675 1 7749 


7823 




6 


7898 


7972 : 8046 


8120 i 8194 


8268 8342 


8416 


8490 


8564 


74 


7 


8638 


8712 8786 


8860 


8934 


9008 9082 


9156 


9230 


9303 




8 


9377 


9451 9525 


9599 


96 73 


9746 9820 


9894 


9968 






















0042 
0778 




9 


770115 


0189 


0263 


0336 


0410 


0484 


0557 


0631 


0705 




590 


0852 


0926 


0999 


1073 


1146 


1220 


1293 


1367 


1440 


1514 




1 


1587 


1661 


1734 


1808 


1881 


1955 


2028 


2102 


2175 


2248 




2 


2322 


2395 


2468 


2542 


2615 


2688 


2762 


2835 


2908 


2981 




3 


3055 


3128 


3201 


3274 


3348 


3421 


3494 


3567 


3640 


3713 




4 


3786 


3860 


3933 


4006 


4079 


4152 


4225 


4298 


4371 


4444 


73 


5 


4517 


4590 j 4663 


4736 


4809 


4882 


4955 


5028 


5100 


5173 




6 


5246 


5319 1 5392 


5465 


5538 


5610 


5683 


5756 


5829 


5902 




7 


5974 


6047 6120 


6193 6265 


6333 


6411 


6483 


6556 


6629 




8 


6701 


6774 6846 


6919 | 6992 


7064 


7137 


7209 7282 


7354 




9 


7427 


7499 , 7572 


7644 7717 


7789 


7862 


7934 


8006 


8079 




600 


8151 


8224 ! 8296 


8368 &441 


8513 


8585 


8658 


8730 


8802 




1 


8874 


8947 1 9019 


9091 9163 


9236 


9308 


9380 


9452 


9524 




2 


9596 


9669 i 9741 


9813 9885 


9957 




















0029 
0749 


0101 A1 ~° 


9245 
0965 




3 


780317 


0389 


0461 


0533 0605 


0677 


0821 


0893 


72 


4 


1037 


1109 


1181 


1253 


1324 


1396 


1468 


1540 


1612 


1684 




5 


1755 


1827 


1899 


1971 


2042 


2114 


2186 


2258 


2329 


2401 




6 


2473 


2544 


2616 


2688 


2759 


2831 


2902 


2974 


3046 


3117 




7 


3189 


3260 3332 


3403 


3475 


3546 


3618 


3689 


3761 


3832 




8 


3904 


3975 


4046 


4118 


4189 


4261 


4332 


4403 


4475 


4546 




9 


4617 


4689 


4760 


4831 


4902 


4974 


5045 


5116 


5187 


5259 




610 


5330 


5401 


5472 


5543 


5615 


5686 


5757 


5828 


5899 


5970 




1 


6041 


6112 


6183 


6254 


6325 


6396 


6467 


6538 


6609 


6680 


71 


2 


6751 


6822 


6893 


6964 


7035 


7106 


7177 


7248 


7319 


7390 




3 


7460 


7531 7602 


7673 7744 


7815 


7885 


7956 


8027 


8098 




4 


8168 


8239 8310 


8381 


8451 


8522 


8593 


8663 ! 8734 


8804 




5 


8875 


8946 | 9016 


9087 


9157 


9228 


9299 


9369 9440 


9510 




6 


9581 


9651 9722 


9792 


9863 


9933 




1 


















0004 
0707 


0074 ni A A 


0215 
0918 




7 


790285 


0356 I 0426 


0496 


0567 


0637 


0778 


0848 




8 


0988 


1059 1129 


1199 


1269 


1340 


1410 


1480 


1550 


1620 




9 


1691 


1761 1831 


1901 


1971 


2041 


2111 


2181 


2252 


2322 




620 


2392 


2462 ! 2532 


2602 


2672 


2742 


2812 


2882 


2952 


3022 


70 


1 


3092 


3162 3231 


3301 


3371 


3441 


3511 


3581 3651 


3721 




2 


3790 


3860 3930 


4000 


4070 


4139 


4209 


4279 1 4349 


4418 




3 


4488 


4558 I 4627 


4697 


4767 


4836 


4906 


4976 | 5045 


5115 




4 


5185 


5254 5324 


5393 5463 


5532 


5602 


5672 5741 


5811 




5 


5880 


5949 6019 


6088 I 6158 


6227 


6297 


6366 * 6436 


6505 




6 


6574 


6644 


6713 


6782 6852 


6921 


6990 


7060 ! 7129 


7198 




7 


7268 


7337 


7406 


7475 


7545 


7614 


7683 


7752 ; 7821 


7890 




8 


7960 


8029 


8098 


8167 


8236 


8305 


8374 


8443 J 8513 


8582 




9 


8651 


8720 


8789 


8858 


8927 

1 


8996 


9065 


9134 9203 


9272 


69 



Proportional Parts. 



Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


75 


7.5 


15.0 


22.5 


30.0 


37.5 


45.0 


52.5 


60.0 


67.5 


74 


7.4 


14.8 


22.2 


29.6 


37.0 


44.4 


51.8 


59.2 


66.6 


73 


7.3 


14.6 


21.9 


29.2 


36.5 


43.8 


51.1 


58.4 


65.7 


72 


7.2 


14.4 


21.6 


28.8 


36.0 


43.2 


50.4 


57.6 


64.8 


71 


7.1 


14.2 


21.3 


28.4 


35.5 


42.6 


49.7 


56.8 


63.9 


70 


. 7.0 


14.0 


21.0 


28.0 


35.0 


42.0 


49.0 


56.0 


63.0 


69 


6.9 


13.8 


20.7 


27.6 


34.5 


41.4 


48.3 


55.2 


62.1 



156 



LOGARITHMS OF NUMBERS. 



No. 630 L. 799.] 










[No. 674 


L. 829. 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 


630 

1 


799341 


9409 


9478 


9547 


9616 


9685 


9754 


9823 


9892 


9961 




800029 


0098 


0167 


0236 


0305 


0373 


0442 


0511 


0580 


0648 


2 


0117 


0786 


0854 


0923 


0992 


1061 


1129 


1198 


1266 


1335 




3 


1404 


1472 


1541 


1609 


1678 


1747 


1815 


1884 


1952 


2021 




4 


2089 


2158 


2226 


2295 


2363 


2432 


2500 


2568 


2637 


2705 




5 


2774 


2842 


2910 


2979 


3047 


3116 


3184 


3252 


3321 


3389 




6 


3457 


3525 


3594 


3662 


3730 


3798 


3867 


3935 


4003 


4071 




7 


4139 


4208 


4276 


4344 


4412 


4480 


4548 


4616 


4685 


4753 




8 


4821 


4889 


4957 


5025 


5093 


5161 


5229 


5297 


5365 


5433 


68 


9 


5501 


5569 


5637 


5705 


5773 


j 5841 


5908 


5976 


6044 


6112 




640 


806180 


6248 


6316 


6384 


6451 


6519 


6587 


6655 


6723 


6790 




1 


6858 


6926 


6994 


7061 


7129 


'• 7197 


7264 


7332 


7400 


7467 




2 


7535 


7603 


7670 


7738 


7806 


7873 


7941 


8008 


8076 


8143 




3 


8211 


8279 


8346 


8414 


8481 


8549 


8616 


8684 


8751 


8818 




4 


8886 


8953 


9021 


9088 


9156 


9223 


9290 


9358 


9425 


9492 




5 


9560 


9627 


9694 


9762 


9829 


9896 


9964 














0031 
0703 


0098 
0770 


0165 
0837 




6 


810233 


0300 


0367 


0434 


0501 


0569 


0636 




7 


0904 


0971 


1039 


1106 


1173 


1240 


1307 


1374 


1441 


1508 


67 


8 


1575 


1642 


1709 


1776 


1843 


1910 


1977 


2044 


2111 


2178 




9 


2245 


2312 


2379 


2445 


2512 


2579 


2646 


2713 


2780 


2847 




650 


2913 


2980 


3047 


3114 


3181 


3247 


a3i4 


3381 


3448 


3514 




1 


3581 


3648 


3714 


3781 


3848 


3914 


3981 


4048 


4114 


4181 




2 


4248 


4314 


4381 


4447 


4514 


4581 


4647 


4714 


4780 


4847 




3 


4913 


4980 


5046 


5113 


5179 


5246 


5312 


5378 


5445 


5511 




4 


5578 


5644 


5711 


5777 


5843 


5910 


5976 


6042 


6109 


6175 




5 


6241 


6308 


6374 


6440 


6506 


6573 


6639 


6705 


6771 


6838 




6 


6904 


6970 


7036 


7102 


7169 


7235 


7301 


7367 


7433 


7499 




7 


7565 


7631 


7698 


7764 


7830 


7896 


7962 


8028 


8094 


8160 




8 


8226 


8292 


8358 


8424 


8490 


8556 


8622 


8688 


8754 


8820 


66 


9 
660 


8885 
9544 


8951 
9610 


9017 
9676 


9083 
9741 


9149 
9807 


{ 9215 
1 9873 


9281 
9939 


9346 


9412 


9478 
























0004 

0661 


C070 
0727 


0136 
0792 




1 


820201 


0267 


0333 


0399 


0464 


0530 


0595 




2 


0858 


0924 


0989 


1055 


1120 


1186 


1251 


1317 


1382 


1448 




3 


1514 


1579 


1645 


1710 


1775 


1841 


1906 


1972 


2037 


2103 




4 


2168 


2233 


2299 


2364 


2430 


2495 


2560 


2626 


2691 


2756 




5 


2822 


2887 


2952 


3018 


3083 


3148 


3213 


3279 


3344 


3409 




6 


3474 


3539 


3605 


3670 


3735 


3800 


3865 


3930 


3996 


4061 




7 


4126 


4191 


4256 


4321 


4386 


4451 


4516 


4581 


4646 


4711 


65 


8 


4776 


4841 


4906 


4971 


5036 


5101 


5166 


5231 


5296 


5361 


9 


5426 


5491 


5556 


5621 


5686 


5751 


5815 


5880 


5945 


6010 




670 


6075 


6140 


6204 


6269 


6334 


6399 


6464 


6528 


6593 


6658 




1 


6723 


6787 


6852 


6917 


6981 


7046 


7111 


7175 


7240 


7305 




2 


7369 


7434 


7499 


7563 


7628 


7692 


7757 


7821 


7886 


7951 




3 


8015 


8080 


8144 


8209 


8273 


1 8338 


8402 


8467 


8531 


8595 




4 


8660 


8724 


8789 


8853 


8918 


i 8982 


9016 


9111 


9175 


9239 




Proportional Parts. 


Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


68 


6.8 


13.6 


20 


4 




27.2 


34.0 


40. g 




47 


.0 




54.4 


61.2 


G7 


6.7 


13.4 


20 


1 




26.8 


33.5 


40.2 




46 


.9 




53.6 


60.3 


66 


6.6 


13.2 


19 


8 




26.4 


33.0 


39.(3 




46 


.2 




52.8 


59.4 


65 


6.5 


13.0 


19 


5 




26.0 


32.5 


39.0 




45 


.5 




52.0 


58.5 


64 


6.4 


12.8 


19 


2 




25.6 


32.0 


33.4 




44 


.8 




51.2 


'57.6 



157 



LOGARITHMS OF NUMBERS. 



No. 


675 L. 829.] 














[N 


o. 719 L. 857. 


N. 





1 2 


3 i 4 

I 


5 


6 


7 


8 


9 


Diff. 


675 


829304 


9368 j 9432 


9497 


9561 


! 9625 


9690 


9754 


9818 


9882 




6 


9947 


! 


















0011 ™~~ 


0139 'wn 


0268 
0909 


0332 ^qo« 


0460 
1102 


0525 
1166 




7 


830589 


0653 


0717 


0781 


0845 


0973 


1037 




8 


1230 


1294 


1358 


1422 


1486 


1550 


1614 


1678 


1742 


1806 


64 


9 


1870 


1934 


1998 


2062 


2126 


2189 


2253 


2317 


2381 


2445 




680 


2509 


2573 


2637 


2700 


2764 


I 2828 


2892 


2956 


3020 


3083 




1 


3147 


3211 


3275 


3338 


3402 


3466 


3530 


3593 


3657 


3721 




2 


3784 


3848 


3912 


3975 


4039 


4103 


4166 


4230 


4294 


4357 




3 


4421 


4484 


4548 


4611 


4675 


4739 


4802 


4866 


4929 


4993 




4 


5056 


5120 


5183 


5247 


5310 


5373 


5437 


5500 


5564 


5627 




5 


5691 


5754 


5817 


5881 


5944 


! 6007 


6071 


6134 


6197 


6261 




6 


6324 


6387 


6451 


6514 


6577 


6641 


6704 


6767 


6830 


6894 




7 


6957 


7020 


7083 


7146 


7210 


7273 


7336 


7399 


7462 


7525 




8 


7588 


7o52 


7715 


7778 


7841 


7904 


7967 


8030 


8093 


8156 




9 


8219 


8282 


8345 


8408 


8471 


; 8534 


8597 


8660 


8723 ! 8786 


63 


690 


8849 


8912 


8975 


9038 


9101 


! 9164 


9227 


9289 


9352 9415 




1 


9478 


9541 


9604 


9667 


9729 


! 9792 


9855 


OOiq i QOQi ' 




1 nrua 




2 


810106 


0169 


0232 


0294 


0357 


j 0420 


0482 


0545 


0608 


0671 




3 


0733 


0796 


0859 


0921 


0984 


, 1046 


1109 


1172 


1234 


1297 




4 


1359 


1422 


1485 


1547 


1610 


1672 


1735 


1797 


1860 


1922 




5 


1985 


2047 


2110 


2172 


2235 


2297 


2360 


2422 


2484 


2547 




C 


2609 


2672 


2734 


2796 


2859 


2921 


2983 


3046 


3108 


3170 




7 


3233 


3295 


3357 


3420 


3482 


3544 


3606 


3669 


3731 


3793 




8 


3855 


3918 


3980 


4042 


'4104 


4166 


4229 


4291 


4353 


4415 




9 


4477 


4539 


4601 


4664 


4726 


4788 


4850 


4912 


4974 


5036 




700 


5098 


5160 


5222 


5284 


5346 


5408 


5470 


5532 


5594 


5656 


62 


1 


5718 


5780 


5842 


5904 


5966 


6028 


6090 


6151 


6213 


6275 




2 


6337 


6399 


6461 


6523 


6585 j 


6646 


6708 


6770 


6832 


6894 




3 


6955 


7017 


7079 


7141 


7202 ' 


7264 


7326 


7388 


7449 


7511 




4 


7573 


7634 


7696 


7758 


7819 


7881 


7943 


8004 


8066 


8128 




5 


8189 


8251 


8312 


8374 


8435 


8497 


8559 


8620 1 8682 


8743 




6 


8805 


8866 


8928 


8989 9051 


9112 


9174 


9235 ! 9297 


9358 




7 


9419 


9481 


9542 


9604 9065 


9726 


9788 


9849 | 9911 


9972 




8 


850033 


0095 


0156 


0217 0279 


0340 : 0401 


0462 0524 


0585 




9 


0046 


0707 


0769 


0830 


0691 


0952 1014 


1075 | 1136 


1197 




710 


1258 


1320 


1381 


1442 


1503 


1564 ! 1625 


1686 ! 1747 


1809 




1 


1870 


1931 


1992 


2053 


2114 


2175 | 2236 


2297 2358 


2419 


61 


2 


2480 


2541 


2602 


2663 


2724 


2785 | 2846 


2907 2968 


3029 


3 


8090 


3150 


3211 3272 


3333 


3394 3455 3516 3577 


3637 




4 


3698 


3759 3820 \ 3881 


3941 


4002 ! 4063 


4124 4185 


4245 




5 


4306 


4367 ! 4428 4488 


4549 


4610 i 4670 


4731 4792 


4852 




6 


4913 


4974 i 5034 


5095 


5156 


5216 i 5277 


5337 5398 


5459 




7 


5519 


5580 


5640 


5701 


5761 


5822 ! 5882 


5943 6003 


6064 




8 


6124 


6185 


6245 


6306 


6366 


6427 1 6487 


6548 6608 


6668 




9 


6729 


6789 


6850 


6910 


6970 


7031 


7091 


7152 


7212 


7272 





Proportional Parts. 



Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


65 


6.5 


13.0 


19.5 


26.0 


32.5 


39.0 


45.5 


52.0 


58.5 


64 


6.4 


12.8 


19.2 


25.6 


32.0 


38.4 


44.8 


51.2 


57.6 


63 


6.3 


12.6 


18 9 


25.2 


31.5 


37.8 


44.1 


50.4 


56.7 


62 


6.2 


12.4 


18.6 


24.8 


31.0 


37.2 


43.4 


49.6 


55.8 


61 


6.1 


12.2 


18.3 


24.4 


30.5 


36.6 


42.7 


48.8 


54.9 


60 


6.0 


12.0 


18.0 


24.0 


30.0 


36.0 1 


42.0 


48.0 


54.0 



158 



ZOGARITHMS OF NUMBERS. 



No. 720 L. 857.] 


[No. 764 L. 883. 


N. 


j 1 | 2 3 

! I i 


4 6 


6 


7 


8 


9 


Diff. 


720 


857332 7393 7453 7513 7574 76&4 7694 


7755 


7815 


7875 




1 


7935 7995 8056 


8116 8176 8236 


8297 


8357 


8417 


8477 




2 


8537 B597 8-357 


8718 1 8778 : 8838 


8898 


8958 


9018 


9078 




3 


9138 9193 9258 


9318 | 9379 9439 


9499 


9559 


9619 


9679 


60 


4 


Q7QQ i Q7QQ Qv."iQ 


(iO-ltf 1 CH\~Q 










Vioa ! otoa aCOa t/t/j.u i/t/iw 

J038 nnoQ 


0158 


0^18 


0278 




5 


860338 0398 0458 0518 0578 0637 


0697 


0757 


0817 


0877 




6 


0937 0996 | 1056 


1116 1176 1236 


1295 


1355 


1415 


1^75 




7 


15:34 1594 1654 


1714 1773 1833 


1393 


1952 


2012 


2072 




8 


2131 2191 2251 


2310 2370 2430 


2439 


2549 


2608 


2668 




9 


2728 ' 2737 j 2847 


2908 2966 3025 


3085 


3114 


3.204 


3263 




730 


3323 3332 ' 3442 


3501 3561 3620 


3680 


3739 


3799 


3858 




1 


3j17 3977 4036 


4096 


4155 4214 


4274 


4333 


4392 


4452 




2 


4511 4570 4630 


4689 


4748 4808 


4^67 


4926 


4985 


5045 




3 


5104 5163 5222 


5232 


5341 ■■ 5400 


5459 


5519 5578 5637 




4 


5696 5755 5S14 


5874 


5933 5992 


6051 


6110 6169 6228 




5 


6287 6346 6405 


6465 


6524 


i 6583 


C642 


6701 6760 6319 


59 


6 


6878 6937 6996 


7055 7114 


7173 


7232 


7291 7350 7409 


7 


74-37 7526 7585 


7644 7703 


7762 


7821 


7939 7998 




8 8056 8115 


8174 


8233 8292 


8350 


8409 


8468 8527 8536 




9 8644 8703 


8762 


8821 ( 8879 


893S 


8997 


9056 9114 91,3 




740 


9232 9290 


9349 


9408 9466 


9525 


9534 


9642 9701 9760 




1 


ORlfl Qfl — ') ; )'-!t f )0 1 * ' 






fkW5 


0170 
0755 


0228 0237 0:345 
0313 0872 0930 




2 


870404 0462 0521 0579 


06:38 0696 




3 


0989 1047 1100 i 1164 


1223 1231 


1339 


1398 1456 1515 




4 


1573 1631 1690 1748 


1306 1865 


1923 


1981 2040 2098 




5 


2156 2215 2273 2331 


2389 2448 


2506 


2564 2622 2681 




6 


2739 2797 2S55 2913 


3030 


3088 


3146 3204 3262 




7 


3331 3379 3437 3495 


3553 3611 


3669 


3727 3735 3344 




8 


3960 4018 4076 


41:34 4192 


42.50 


4303 4366 4424 


58 


9 


4482 4540 , 4598 


4656 


4714 4772 


4830 


4388 ; 4945 


5003 




750 


5061 5119 5177 


5235 


5293 5351 


5409 


5466 5524 


5582 




1 


5640 5698 5756 


5313 


5^71 5929 


5987 


6045 6102 


6160 




8 


6218 6276 6333 


6391 


6449 6507 


6564 


6622 6630 


6737 




3 


6795 6853 6910 


6968 


7026 7083 


7141 


7199 7256 


7314 




4 


7371 7429 i 7487 


7544 


7602 7659 


7717 


7774 7832 


7889 




5 


7947 8004 


8119 


8177 8234 


8292 


8-349 8407 


8464 




6 


8522 8579 8637 


8694 


8752 8809 


8866 


8924 8931 


9039 




7 


9096 9153 9211 


9268 


9325 9383 


9440 


9497 9555 


9612 




3 


QAAQ Q~Oii 0?<1 


9841 


99Sfi 












0013 


0070 , 0127 
0642 0699 


0185 
0756 




9 880242 


0299 . 0:356 0413 


0471 0528 0585 




7G0 


0314 


0371 09-23 0935 


1042 1099 


1156 


1213 1271 


1328 




1 


1385 


1442 


1499 1556 


1613 1670 


1727 


1734 1841 


1893 


57 


2 


1955 


2012 


20,39 2126 21<3 2240 


2297 


23.54 2411 


2468 


3 


2525 


2581 


2638 2695 


2752 2809 


2866 


2923 I 2980 


3037 




4 


3093 


3150 


3007 3264 


3321 3377 


3434 


3491 3543 


3605 




Proportional Parts. 


Diff. 


1 


2 3 4 


5 6 


7 


8 


9 


59 


5.9 11.8 


17.7 


23.6 


29.5 35.4 




41.3 


47.2 


53.1 


58 


5.8 S 11.6 


17.4 


23.2 


29.0 34.8 




40.6 


4u.4 


52.2 


57 


5.7 j 11.4 


17.1 


22.8 


28.5 34.2 


39.9 


45.6 


51.3 


56 


5.6 11.2 


16.8 


22.4 


28.0 SJ.6 


39.2 1 44.3 


50.4 



159 



LOGARITHMS OF NUMBERS. 



No. 765 L. 883.] 



[No. 809 L. ! 



N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 


765 


883661 


3718 


3775 


3832 


3888 


3945 


4002 


4059 


4115 


4172 




6 


4229 


4285 


4342 


4399 


4455 


4512 


4569 


4625 


4682 


4739 




7 


4^95 


4852 


4909 


4965 


5022 


5078 


5135 


5192 


5248 


5305 




8 


5361 


5418 


5474 


5531 


5587 


5644 


5700 


5757 


5813 


5870 




9 


5926 


5983 


6039 


6096 


6152 


6209 


6265 


6321 


6378 


6434 




770 


6491 


6547 


6604 


6660 


6716 


6773 


6829 


6885 


6942 


6998 




1 


7054 


7111 


7167 


7223 


7280 


7336 


7392 


7449 


7505 


7561 




2 


7617 


7674 


7730 


7786 


7842 


7898 


7955 


8011 


8067 


8123 




3 


8179 


8236 


8292 


8348 


8404 


8460 


8516 


8573 


8629 


8685 




4 


8741 


8797 


8853 


8909 


8965 


! 9021 


9077 


9134 


9190 


9246 




5 


9302 


9358 


9414 


9470 


9526 


i 9582 


9638 


9694 


9750 


9806 


56 


6 


9862 


9918 


9974 


















0030 
0589 


0086 


0141 


0197 
0756 


0253 

0812 


0309 
0868 


0365 
0924 




7 


890421 


0477 


0533 


0645 


0700 




8 


0980 


1035 


1091 


1147 


1203 


1259 


1314 


1370 


1426 


1482 




9 


1537 


1593 


1649 


1705 


1760 


1816 


1872 


1928 


1983 


2039 




780 


2095 


2150 


2206 


2262 


2317 


2373 


2429 


2484 


2540 


2595 




1 


2651 


2707 


2762 


2818 


2873 


2929 


2985 


3040 


3096 


3151 




2 


3207 


3262 


3318 


3373 


&429 


3484 


3540 


3595 


3651 


3706 




3 


3762 


3817 


3873 


3928 


3984 


; 4039 


4094 


4150 


4205 


4261 




4 


4316 


4371 


4427 


4482 


4538 


' 4593 


4648 


4704 


4759 


4814 




5 


4870 


4925 


4980 


5036 


5091 


5146 


5201 


5257 


5312 


5367 




6 


5423 


5478 


5533 


5588 


5644 


5699 


5754 


5809 


5864 


5920 




7 


5975 


6030 


6085 


6140 


6195 


6251 


6306 


6361 


6416 


6471 




8 


6526 


6581 


GG36 


6692 


6747 


i 6802 


6857 


6912 


6967 


7022 




9 


7077 


7132 


7187 


7242 


7297 


7352 


7407 


7462 


7517 


7572 


55 


790 


7627 


7682 


7737 


7792 


7847 


7902 


7957 


8012 


8067 


8122 


1 


8176 


8231 


8286 


8&41 


8396 


8451 


8506 


8561 


8615 


8670 




2 


8725 


8780 


8835 


8890 


8944 


8999 


9054 


9109 


9164 


9218 




3 


9273 


9328 


9383 


9437 


9492 


j 9547 


9602 


9656 


9711 


9766 




4 


9821 


9875 


9930 


9985 
















0039 


0094 
0640 


0149 
0695 


0203 
0749 


0258 
0804 


0312 
0859 




5 


900367 


0422 


0476 


0531 


0586 




G 


0913 


0968 


1022 


1077 


1131 


, 1186 


1240 


1295 


1349 


1404 




7 


1458 


1513 


1567 


1622 


1676 


1731 


1785 


1840 


1894 


1948 




8 


2003 


2057 


2112 


21G6 


2221 


i 2275 


2329 


2384 


2438 


2492 




9 


2547 


2601 


2G55 


2710 


27G4 


2818 


2873 


2927 


2981 


3036 




800 


3090 


3144 


3199 


3253 


3307 


3361 


3416 


3470 


3524 


3578 




1 


3633 


3687 


3741 


3795 


3849 


1 3904 


3958 


4012 


4066 


4120 




2 


4174 


4229 


4283 


4337 


4391 


■ 4445 


4499 


4553 


4607 


4661 




3 


4716 


4770 


4824 


4878 


4932 


i 4986 


5040 


5094 


5148 


5202 


54 


4 


5256 


5310 


53G4 


5418 


5472 


! 5526 


5580 


5634 


5688 


5742 


5 


5796 


5850 


5904 


5958 


6012 


6066 


6119 


6173 


6227 


6281 




6 


6335 


6389 


6443 


6497 


6551 


6604 


6658 


6712 


6766 


6820 




7 


6874 


6927 


6981 


7035 


7089 


7143 


7196 


7250 


7304 


7358 




8 


7411 


7465 


7519 


7573 


7626 


7680 


7734 


7787 


7841 


7895 




9 


7949 


8002 


8056 


8110 


8163 


8217 


8270 


8324 


8378 


8431 





Proportional Parts. 



Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


57 
56 
55 
54 


5.7 
5.6 
5.5 
5.4 


11.4 
11.2 
11.0 

10.8 


17.1 
16.8 
16.5 
16.2 


22.8 
22.4 
22.0 
21.6 


28.5 
28.0 
27.5 
27.0 


34.2 
33.6 
33.0 
32.4 


39.9 
39.2 

38.5 
37.8 


45.6 
44.8 

44.0 
43.2 


51.3 
50.4 
49.5 
48.6 



160 



LOGARITHMS OF NUMBERS. 



No. 810 L. 908.] 



[No. S54 L. 931. 



N. 










S10 908485 
1 9021 

2 



9556 



3 

4 
5 
6 

r 

8 
9 

820 

1 



3 

4 
5 
6 
7 
8 
9 

&40 
1 
2 
3 
4 
5 
6 
7 
& 
9 

850 

1 



8539 i 8592 

9128 

9610 9663 



8646 
9181 
9716 



8699 8753 
9285 9289 
9770 9823 



910091 
0624 
1158 
1690 
2222 
2753 
3284 

3814 
4343 
4872 
5400 
5927 
64.54 
6980 
7506 
8030 
8555 

9078 
9601 



0144 0197 

0678 0731 

1211 1264 

1743 1797 

2275 2328 

2806 2859 



920123 
0645 
1166 
1686 
2206 
2725 
3244 
3762 

4279 
4796 
5312 
5828 
6342 
6857 
7370 
7883 
8396 
8908 

9419 



3337 

3867 
4396 
4925 
5453 
5980 
6507 
7033 
7558 
8083 
8607 

9130 
9653 



3390 

3920 
4449 
4977 
5505 
6033 
6559 
7085 
7611 
8135 
8659 

9183 
9706 



0251 
0784 
1317 
1850 
2381 
2913 
3443 

3973 
4502 
5030 
5558 
6085 
6612 
7138 
7663 
8188 
8712 

9235 
9758 



0176 
0697 
1218 
17:38 



0228 
0749 
1270 
1790 



930440 
0949 
1458 



2258 2310 
2777 



0304 
0838 
1371 
1903 
2435 
2966 
3496 

4026 
4555 
5083 
5611 
6138 
6664 
7190 
7716 
8240 
8764 



0:358 
0891 
1424 
1956 
2488 
3-»19 
S549 

4079 
4608 
5136 
5664 
6191 
6717 
7243 

8293 
8816 



9287 9340 
9810 0862 



0280 0:332 

0801 (853 

1322 1374 

1842 1894 



3296 
3814 



3:348 
3865 



4331 4383 
4848 4899 



5364 
5879 
6394 
6906 
7422 
7935 
8447 
6959 



6415 
5931 
6445 
6959 
7473 
7986 
8498 
9010 



236; 
2881 
3399 
3917 



2414 
2933 
3451 
3969 



4434 4486 

4951 5003 



5518 
6034 
6.548 
7062 
7576 



9470 f 9521 
9961 



0032 
0491 I 0.542 
1000 I 1051 
1509 ; 1560 



546, 

5982 

6497 

7011 

7524 

8037 8088 

8549 8601 

9061 9112 

9572 9623 

0083 I 0134 

0592 0643 

1102 ; 1153 

1610 1661 



0384 
0906 
1426 
1946 
2466 

^5(3 

4538 
5054 
5570 



0185 
0694 
1204 
1712 



8807 
9342 



9896 
9930 



0411 
0944 
1477 
2009 
2541 
3072 
3602 | 

4132 

4660 
5180 ) 
5716 
6243 
6770 j 
7295 
7820 ! 
8345 
6869 i 

9392 I 
9914 



0464 
0998 
1530 
2063 
2594 
3125 
3655 

4184 
4713 
5241 

5769 
6296 
(822 
7348 
7873 
8397 
8921 

9444 
9967 



8914 8967 
9449 9503 

9984 

0037 

0518 0571 
1051 I 1104 
1584 1C37 
2116 2169 
2647 ' 2700 
3178 , 3231 
3708 3761 



4237 
4766 
5294 
5822 

6349 
I 6875 

7400 
i 7925 

8450 
i 8973 

9406 



4290 
4819 
5347 
5875 
6401 
6927 
7453 
7078 
8502 
9026 

9549 



0436 

1478 
1998 
2518 
3037 

4072 

4589 
5106 
?.621 



6085 


6137 


6600 


6651 


7114 


7165 


7627 


7678 


8140 


8191 


8652 


8703 


9163 


9215 


9674 


97'25 



0489 

1010 
1530 
2050 
2570 
3089 
S607 
4124 

4641 

5157 
5673 
6188 
6702 
7216 
7730 
8242 
6754 
9266 



C019 
(541 
1C62 
1562 
2102 
2C22 



0071 
C503 
1114 
1634 
2154 
2674 



S140 3192 
£058 S710 
4176 4228 



4693 
E209 

57-5 
6240 
6754 
7268 
7781 
8293 

eeos 

9317 



4744 
52G1 
5776 
6291 

6^05 
7219 

res2 

8345 
8857 

SoCS 

9776 0827 0679 



0236 0287 

0745 0706 

1254 1305 

1763 1814 



0S38 0389 
0847 : 0898 
1356 1407 
1865 1915 



Diff. 



Proportional Parts. 



Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


53 


5.3 


10.6 


15.9 


21.2 


26.5 


31.8 


37.1 


42.4 


47.7 


52 


5.2 


10.4 


15.6 


20.8 


26.0 


31.2 


36.4 


41.6 


46.8 


51 


5.1 


10.2 


15.3 


20.4 


25.5 


30.6 


35.7 


40.8 


45.9 


50 


5.0 


10.0 


15.0 


20.0 


25.0 


30.0 


35.0 


40.0 


45.0 



161 



LOGARITHMS OF NUMBERS. 



No. 


855 L. 931.1 














[Xo. 899 L. 954. 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 


855 


931986 


2017 


2068 


2118 


2169 


! 2220 


2271 


2322 


2372 


2423 




6 


2474 


2524 


2575 


2626 


: 2677 


2737 


2778 


2829 


2879 


2930 




7 


2981 


3031 


3082 


3133 


3183 


3234 


3285 


3335 


3386 


3437 




8 


3487 


3538 


3589 


3639 


3690 


3740 


3791 


3841 


3S92 


3943 




9 


3993 


4044 


4094 


4145 


4195 


4246 


4296 


4347 


4397 


4448 




860 


4498 


4549 


4599 


4650 


4700 


47'51 


4801 


4852 


4902 


4953 




1 


5003 


5054 


5104 


5154 


5205 


5255 


5306 


5356 


5406 


5457 




2 


5507 


5558 


5608 


5658 


5709 


5759 


5809 


5860 


5910 


5960 




3 


6011 


6061 


6111 


6162 


6212 


6262 


6313 


6363 


6413 


6463 




4 


6514 


6564 


6614 


6665 


6715 


6765 


6815 


6865 


6916 


6966 




5 


7016 


7066 


7116 


7167 


7217 


7267 


7317 


7367 


7418 


7468 




6 


7518 


7568 


7618 


7668 


7718 


i 7769 


7819 


7869 


7919 


7969 


50 


7 


8019 


8069 


8119 


8169 


8219 


82(39 


8520 8370 


8420 


8470 


8 


8520 


8570 


8620 


8670 


8720 


! 8770 


8820 J 8870 


8920 


8970 




9 


9020 


9070 


9120 


9170 


9220 


9270 


9320 j 9369 


9419 


9469 




870 


9519 


9569 


9619 


9669 


9719 


9769 


9819 ! 9S69 


«918 


9968 




1 


940018 


0068 


0118 


0168 


0218 


(T267 


0317 i 0367 


0417 


0467 




2 


0516 


0566 


0616 


0666 


0716 


i 0765 


0815 


0865 


0915 


0964 




3 


1014 


1064 


1114 


1163 


1213 


1263 


1313 


1362 


1412 


1462 




4 


1511 


1561 


1611 


1660 


1710 


1760 


1809 


1859 


1909 


1958 




5 


2008 


2058 


2107 


2157 


2207 


2256 


2306 2355 


2405 2455 




6 


2504 


2o54 


2603 


2653 


2702 


2752 


2801 2851 


2901 2950 




7 


3000 


3049 


3099 


3148 


3198 


3247 


3297 3346 


3396 


3445 




8 


3495 


3544 


3593 


3643 


3692 


3742 


3791 3841 


3890 


3939 




9 


3989 


4038 


408S 


4137 


4186 


1 4236 


4285 j 4335 


4384 


4433 




880 


4483 


4532 


4581 


4631 


4680 


4729 


4779 4828 


4877 


4927 




1 


4976 


5025 


5074 


5124 


5173 


5222 


5272 5321 


5370 


5419 




2 


5469 


5518 


5567 


5616 


5665 


5715 


5764 ! 5813 


5862 


5912 




3 


5961 


6010 


6059 


6108 


6157 


' 6207 


6256 6305 


6354 


6403 




4 


6452 


6501 


6551 


6600 


6649 


: 6698 


6747 6796 


6845 


6894 




5 


6943 


6992 


7041 


7090 


7139 


7189 


7238 ; 7287 


7336 


7385 


49 


6 


7434 


7483 


7532 


7581 


7630 


7679 


7r28 77'77 


7826 


7875 


7 


7924 


7973 


8022 


8070 


8119 


8168 


8217 8266 


8315 


8364 




8 


8413 


8462 


8511 


8560 


8608 


8657 


8706 | 8755 


8804 


8853 




9 


8902 


8951 


8999 


9048 


9097 


9146 


9195 9244 


9292 


9341 




890 


9390 


9439 


9488 


9536 


9585 


9634 


9683 9731 


9780 


9829 




1 


9878 


9926 


9975 
















0024 
0511 


0073 : 
0560 


0121 
0608 


0170 0219 
0657 0706 


0267 
0754 


0316 

0803 




2 


950365 


0414 


0462 




3 


0851 


0900 


0949 


0997 


1046 l 


1095 


1143 


1192 


1240 


1289 




4 


1338 


1386 


1435 


1483 


1532 


1580 


1629 


1677 


1726 


1775 




5 


1823 


1872 


1920 


1969 


2017 ! 


2066 


2114 


2163 


2211 


2S60 




6 


2308 


2356 


2405 


2453 


2502 


2550 


2599 


2647 


2696 


2744 




7 


2792 


2841 


2889 


2938 


2986 


3034 


3083 


3131 


3180 


3228 




8 


3276 


3325 


3373 


3421 


3470 


3518 


3566 


3615 


3663 


3711 




9 


3760 


3808 


3856 


3905 


3953 


4001 


4049 


4098 


4146 


4194 





Proportional Parts. 



Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


51 
50 
49 
48 


5.1 
5.0 
4.9 

4.8 


10.2 
10.0 
9.8 
9.6 


15.3 
15.0 
14.7 
14.4 


20.4 
20.0 
19.6 
19.2 


25.5 
25.0 
24.5 
24.0 


30.6 
30.0 
29.4 
28.0 


35.7 
35.0 
34.3 
33.6 


40.8 
40.0 
39.2 
38.4 


45.9 
45.0 
44.1 
43.2 



162 



LOGARITHMS OF NUMBERS. 



No 900 L. 954.1 














[>' 


o. 944 L. 975. 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 


900 


954243 


4291 


4339 


4387 


4435 


4484 


4532 


4580 


4628 


4677 




1 


4725 


4773 


4N21 


4869 


4918 


i 4966 


5014 


5062 


5110 


5158 




2 


5207 


5255 


5303 


5351 


5399 


5447 


5495 


5.543 


5592 


5640 




3 


5688 


57:36 


57&4 


5832 


5880 


1 5928 


5976 


6024 


6072 


6120 




4 


6168 


6216 


6265 


6313 


6361 


6409 


6457 


6505 


6553 


6601 


48 


5 


6649 


6697 


6745 


6793 


6840 


i 6888 


6936 


6984 


7032 


7080 


6 


7128 


7176 


7224 


7272 


7320 


! 7368 


7416 


7464 


7512 


7559 




7 


7607 


7655 


7703 


7751 


7799 


7847 


7894 


7942 


7990 


8038 




8 


8086 


8134 


8181 


8229 


8277 


8325 


8373 


8421 


&68 


8516 




9 


8564 


8612 


8659 


8707 


8755 


8803 


8850 


8898 


8946 


8994 




910 


9041 


9089 


9137 


9185 


9232 


9280 


9328 


9375 


9423 


9471 




1 


9518 


9566 


9614 


9661 


9709 


9757 


9804 


9852 


9900 


9947 




2 


9995 






















0042 


0090 


0138 


0185 


0233 


0230 


0328 


0376 


0423 




3 


960471 


0518 


0566 


0613 


0661 


: 0709 


0756 


0804 


0851 


0899 




4 


0946 


0994 


1041 


1089 


1136 


1184 


1231 


1279 


1326 


1374 




5 


14/21 


1469 


1516 


1563 


1611 


1658 


1706 


1753 


1801 


1848 




6 


1895 


1943 


1990 


2038 


2085 


2132 


2180 


2227 


2275 


2322 




7 


2369 


2417 


2464 


2511 


2559 


\ 2606 


2653 


2701 


2748 


2795 




8 


2843 


2890 


2937 


2985 


3032 


; 3079 


3126 


3174 


3221 


3268 




9 


3316 


3363 


3410 


3457 


3504 


3552 


3599 


3646 


3693 


3741 




920 


3788 


3835 


3882 


3929 


3977 


! 4024 


4071 


4118 


4165 


4212 




1 


4260 


4307 


4354 


4401 


4448 


, 4495 


4542 


4590 


4637 


46S4 




2 


4731 


4778 


4825 


4872 


4919 


i 4966 


5013 


5061 


5108 


5155 




3 


5202 


5249 


5296 


5343 


5390 


5437 


54&4 


5531 


5578 


5625 




4 


5672 


5719 


5766 


5813 


5860 


! 5907 


5954 


6001 


6048 


6095 


47 


5 


6142 


6189 


6236 


6283 


6329 


6376 


6423 


6470 


6517 


6564 




6 


6611 


6658 


6705 


6752 


6799 


> 6&45 


6892 


6939 


69S6 


70&3 




7 


7080 


7127 


7173 


7220 


7267 


7314 


7361 


7408 


74.54 


7501 




8 


7548 


7595 


7642 


7688 


7735 


7782 


7829 


7875 


7922 


7969 




9 


8016 


8062 


8109 


8156 


8203 


8249 


8296 


8343 


8390 


8436 




930 


8483 


8530 


8576 


8623 


8670 


1 8716 


6763 


8*10 


8856 


8903 




1 


8950 


8996 


9043 


9090 


9136 


9183 


9229 


9276 


9323 


93G9 




2 


9416 


9463 


9509 


9556 


9602 


9649 


9695 


9742 


9789 


9835 




3 


9882 


9928 


9975 


















0021 
0486 


0068 
0533 


. 0114 

0579 


0161 
0626 


0207 
0672 


0254 


0300 




4 


970347 


0393 


0440 


C719 


0765 




5 


0812 


0858 


0904 


0951 


0997 


1044 


1090 


1137 


1183 


1229 




6 


1276 


1322 


1369 


1415 


1461 


i 1508 


1554 


1601 


1647 


1693 




7 


1740 


1786 


1S32 


1879 


1925 


: 1971 


2018 


2064 


2110 


2157 




8 


2203 


2249 


2295 


2342 


2388 


2434 


2481 


2527 


2573 


2619 




9 


2666 


2712 


2758 


2804 


2851 


2897 


2943 


2989 


3035 


3082 




940 


3128 


3174 


3220 


3266 


3313 


3359 


3405 


3451 


3497 


3543 




1 


3590 


3636 


3682 


3728 


3774 


3820 


3866 


3913 


3959 


4005 




2 


4051 


4097 


4143 


4189 


4235 


4281 


4327 


4374 


4420 


4466 




3 


4512 


4558 


4604 


4650 


4696 


4742 


4788 


4834 


4880 


4926 




4 


4972 


5018 


5064 


5110 


5156 


5202 


5248 


5294 


5340 


5386 


46 








Pro] 


=>ORTIO 


nalPaj 


RTS. 








Diff. 


1 


2 


3 




4 


5 


6 




7 


8 


9 


47 4.7 


9.4 


14 


1 


" 


18.8 


23.5 


28.2 




32 


.9 




37.6 


42.3 


46 4.6 


9.2 


13 


8 


" 


L8.4 


23.0 


27.6 




32 


.2 




36.8 41.4 



163 



LOGARITHMS OF NUMBERS. 



No. 945 L. 975.] 



[No. 989 L. 995. 



N. 





1 


8 


a 


4 


5 


6 


7 


8 


9 


Diff. 


945 


975432 


5478 


5524 


5570 


5616 


5662 


5707 


5753 


5799 


5845 




6 


5891 


£937 


5983 


6029 


6075 


6121 


6167 


6212 


6258 


6304 




7 


6350 


6396 


6442 


6488 


6533 ; 


6579 


6625 


6671 


6717 


6763 




8 


6808 


6854 


6900 


6946 


6992 ! 


7037 


7083 


7129 


7175 


7220 




9 


7266 


7312 


7358 


7403 


7449 


7495 


7541 


7586 


7632 


7678 




950 


7724 


7769 


7815 


7861 


7906 


7952 


7998 


8043 


8089 


8135 




1 


8181 


8226 


8272 


8317 


8363 


8409 


8454 


8500 


8546 


8591 




2 


8637 


8683 


8728 


8774 


8819 


; 8865 


8911 


8956 


9002 


9047 




3 


9093 


9138 


9184 


9230 


9275 


9321 


9366 


9412 


9457 


9503 




4 


9548 


9594 


9639 


9685 


9730 


j 9776 


9821 


9867 


9912 


9958 




5 


980003 


0049 


0094 


0140 


0185 


0231 


0276 


0322 


0367 


0412 




6 


0458 


0503 


0549 


0594 


0640 


0685 


0730 


0776 


0821 


0867 




7 


0912 


0957 


1003 


1048 


1093 


1139 


1184 


1229 


1275 


1320 




8 


1366 


1411 


1456 


1501 


1547 


1592 


1637 


1683 


1728 


1773 




9 


1819 


1864 


1909 


1954 


2000 


2045 


2090 


2135 


2181 


2226 




960 


2271 


2316 


2362 


2407 


2452 


2497 


2543 


2588 


2633 


2678 




1 


2723 


2769 


2814 


2859 


2904 


1 2949 


2994 


3010 


3085 


3130 




2 


3175 


3220 


3265 


3310 


3356 


' 3401 


3446 


3491 


3536 


3581 




3 


3626 


3671 


3716 


3762 


3807 


3852 


3897 


3942 


3987 


4032 




4 


4077 


4122 


4167 


4212 


4257 


4302 


4347 


4392 


4437 


4482 


45 


5 


4527 


4572 


4617 


4662 


4707 


4752 


4797 


4842 


4887 


4932 


6 


4977 


5022 


5067 


5112 


5157 


5202 


5247 


5292 


5337 


5382 




7 


5426 


5471 


5516 


5561 


5606 


5651 


5696 


5741 


5786 


5830 




8 


5875 


5920 


5965 


6010 


6055 


6100 


6144 


6189 


6234 


6279 




9 


6324 


6369 


6413 


6458 


6503 


6548 


6593 


6637 


6682 


6727 




970 


6772 


6817 


6861 


6906 


6951 


6996 


7040 


7085 


7130 


7175 




1 


7219 


7264 


7309 


7353 


7398 


7443 


7488 


7532 


7577 


7622 




2 


7666 


7711 


7756 


7800 


7845 


. 7890 


7934 


7979 


8024 


8068 




3 


8113 


8157 


8202 


8247 


8291 


; 8336 


8331 


8425 


8470 


8514 




4 


8559 


8604 


8648 


8693 


8737 


8782 


8826 


8871 


8916 


8960 




5 


9005 


9049 


9094 


9138 


9183 ; 


9227 


9872 


9316 


9361 


9405 




6 


9450 


9494 


9539 


9583 


9628 


I 9672 


9717 


9761 


9806 


9850 




7 


9895 


9939 


9983 




1 














0028 
0472 


0072 
0516 


1 0117 
0561 


0161 

0605 


0206 


0250 


0294 




8 


990339 


0383 


0428 


0650 


0694 


0738 




9 


0783 


0827 


0871 


0916 


0960 


1004 


1049 


1093 


1137 


1182 




980 


1226 


1270 


1315 


1359 


1403 


1448 


1492 


1536 


1580 


1625 




1 


1669 


1713 


1758 


1802 


1846 


1890 


1935 


1979 


2023 


2067 




2 


2111 


2156 


2200 


2244 


2288 


| 2333 


2377 


2421 


2465 


2509 




3 


2554 


2598 


2&12 


2686 


2730 


: 2774 


2819 


2863 


2907 


2951 




4 


2995 


3039 


3083 


3127 


3172 1 


3216 


3260 


3304 


3348 


3392 




5 


3436 


3480 


3524 


3568 


3613 ' 


3657 


3701 


3745 


3789 


3833 




6 


3877 


3921 


3965 


4009 


4053 


1 4097 


4141 


4185 


4229 


4273 




7 


4317 


4361 


4405 


4449 


4493 


4537 


4581 


4625 


4669 


4713 


44 


8 


4757 


4801 


4845 


4889 


4933 ' 


4977 


5021 


5065 


5108 


5152 




9 


5196 


5240 


5284 


5328 


5372 1 


5416 


5460 


5504 


5547 


5591 





Proportional Parts. 



Diff. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


46 


4.6 


9.2 


13.8 


18.4 


23.0 


27.6 


32.2 


36.8 


41.4 


45 


4.5 


9.0 


13.5 


18.0 


22.5 


27.0 


31.5 


36.0 


40.5 


44 


4.4 


8.8 


13.2 


17.6 


22.0 


26.4 


30.8 


35.2 


39.6 


43 


4.3 


8.6 


12.9 


17.2 


21.5 


25.8 


30.1 


34.4 


38.7 



164 



LOGARITHMS OF NUMBERS, 



No. 


390 L. 995.] 














[No. 999 L. 999. 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Difl. 


090 


995635 


5679 


5723 


5767 


5811 


5854 


5898 


5942 


5986 ' 6030 




1 


6074 


6117 


6161 


6205 


6249 


6293 


6337 


6380 


6424 6468 


44 


2 


6512 


6555 


6599 


6643 


6687 


6731 


6774 


6818 


6862 i 6906 




8 


6949 6993 


7037 


7080 


7124 


7168 


7212 


7255 


7299 7343 




4 


7386 7430 


7474 


7517 


7561 


7605 


7648 


7692 


7736 7779 




5 


7823 


7867 


7910 


7954 


7998 


8041 


8085 


8129 


8172 | 8216 




6 


8259 


8303 


8347 


8390 


8434 


8477 


8521 


8564 


8608 ! 8652 




7 


8095 


8739 


8782 


8826 


8869 


8913 


8956 


9000 


9043 9087 




8 


9131 


9174 


9218 


9261 


■ 9305 


9348 


9392 


9435 


9479 


9522 




9 


9565 


9609 


9652 


9696 


9739 


, 9783 


9826 


9870 


9913 


9957 


43 



Logarithms of Numbers from 1 to 100. 



N. 


Log. 


1 


0.000000 


2 


0.301030 


3 


0.477121 


4 


0.602060 


5 


0.698970 


6 


0.778151 


7 


0.845098 


8 


0.903090 


9 


0.954243 


10 


1.000000 


11 


1.041393 


12 


1.079181 


13 


1.113943 


14 


1.146128 


15 


1.176031 


16 


1.204120 


17 


1.230449 


18 


1.255273 


19 


1.278754 


20 


1.301030 



N. 


Log. 


21 


1.322219 


22 


1.342423 


23 


1.361728 


24 


1.380211 


25 


1.397940 


26 


1.414973 


27 


1.431364 


28 


1.447153 


29 


1.46239S 


30 


1.477121 


31 


1.491362 


32 


1.505150 


33 


1.518514 


34 


1.531479 


35 


1.544068 


36 


1.556302 


37 


1.568202 


38 


1.579784 


39 


1.591065 


40 


1.602060 



N. 


Log. 


41 


1.612784 


42 


1.623249 


43 


1.633468 


44 


1.643453 


45 


1.653213 


46 


1.662758 


47 


1.672098 


4^ 


1.681241 


49 


1.690196 


50 


1.698970 


51 


1.707570 


52 1.716003 


53 1.724276 


54 1 . 732394 


55 1.740363 


56 1.748188 


57 1 . 755875 


58 1.763428 


59 1.770852 


60 


1.778151 



N. 


Log. 


61 


1.785330 


62 


1.792392 


63 


1.799341 


64 


1.806180 


65 


1.812913 


66 


1.819544 


67 


1.826075 


68 


1.832509 


69 


1.838849 


70 


1.845098 


71 


1.851258 


72 


1.857332 ! 


73 


1.863323 


74 


1.869232 


75 


1.875061 


76 


1.880814 


77 


1.886491 


78 


1.892)95 


79 


1.897627 


80 


1.903090 



N. 


Log. 


81 


1.9C8485 


82 


1.913814 


83 


1.919073 


81 


1.924279 


85 


1.929419 


86 


1.934498 


87 


1.939519 


88 


1.944483 


89 


1.949390 


9> 


1.954243 


91 


1.959041 


92 


1.963788 


93 


1.968483 


94 


1.973128 


95 


1.977724 


96 


1.982271 


97 


1.986772 


98 


1.991226 


99 


1.995635 


100 


2.000000 



165 



LOGAKITHMIC SINES, COSINES, 
TANGENTS AND COTANGENTS. 



LOGARITHMIC SINES. 



179° 



If 


i 


Sine. 


q-l 


Tang. 


Cotang. 


q + l 


Dl" 


Cosine. 


f 








4.685 






15.314 




1 








Inf. neg. 


575 


575 


Inf. neg. 


Inf. pos. 


425 




ten 


60 


60 


1 


6.463726 


575 


575 


6.463726 


13.536274 


425 




ten 


59 


120 


2 


.764756 


575 


575 


.764756 


.235244 


425 




ten 


58 


180 


3 


6.940847 


575 


575 


6.940847 


13.059153 


425 




ten 


57 


240 


4 


7.065786 


575 


575 


7.065786 


12.934214 


425 




ten 


56 


300 


5 


.162696 


575 


575 


.162696 


.837304 


425 


.02 

.00 
.00 
.00 
.02 


ten 


55 


360 


6 


.241877 


575 


575 


.241878 


.758122 


425 


9.999999 


54 


420 


7 


.308824 


575 


575 


.308825 


.691175 


425 


.999999 


53 


480 


8 


.366816 


574 


576 


.366817 


.633183 


424 


.999999 


52 


540 


9 


.417968 


574 


576 


.417970 


.582030 


424 


.999999 


51 


600 


10 


.463726 


574 


576 


.463727 


.536273 


424 


.999998 


50 


660 


11 


7.505118 


574 


576 


7.505120 


12.494880 


424 


.00 

.02 
.00 
.02 
.00 
.02 
.00 
.02 
.02 
.00 


9.999998 


49 


720 


12 


.542906 


574 


577 


.542909 


.457091 


423 


.999997 


48 


780 


13 


.577668 


574 


577 


.577672 


.422328 


423 


.999997 


47 


840 


14 


.609853 


574 


577 


.609857 


.390143 


423 


.999996 


46 


900 


15 


.639816 


573 


578 


.639820 


.360180 


422 


.999996 


45 


960 


16 


.667845 


573 


578 


.667849 


.332151 


422 


.999995 


44 


1020 


17 


.694173 


573 


578 


.694179 


.305821 


422 


.999995 


43 


1080 


18 


.718997 


573 


579 


.719003 


.280997 


421 


.999994 


42 


1140 


19 


.742478 


573 


579 


.742484 


.257516 


421 


.999993 


41 


1200 


20 


.764754 


572 


580 


.764761 


.235239 


420 


.999993 


40 


1260 


21 


7.785943 


572 


580 


7.785951 


12.214049 


420 


.C2 
.02 
.02 
.02 
.00 
.02 
.02 
.02 


9.999992 


39 


1320 


22 


.806146 


572 


581 


.806155 


.193845 


419 


.999991 


38 


1380 


23 


.825451 


572 


581 


.825460 


.174540 


419 


.999990 


37 


1440 


24 


.843934 


571 


582 


.843944 


.156056 


418 


.999986 


36 


1500 


25 


.861662 


571 


583 


.861674 


.138326 


417 


.999989 


35 


1560 


26 


.878695 


571 


583 


.878708 


.121292 


417 


.999988 


34 


1620 


27 


.895085 


570 


584 


.895099 


.104901 


f 416 


.999987 


33 


1680 


28 


.910879 


570 


584 


.910894 


.089106 


416 


.999986 


32 


1740 


29 


.926119 


570 


585 


.926134 


.073866 


415 


.02 
.03 


.999985 


31 


1800 


30 


.940842 


569 


586 


.940858 


.059142 


414 


.999983 


30 


1860 


31 


7.955082 


569 


! 587 


7.955100 


12.044900 


413 


.02 
.02 
.02 
.02 
.03 


9.999982 


29 


1920 


32 


.968870 


569 


587 


.968889 


.031111 


413 


.999981 


28 


1980 


33 


.982233 


568 


588 


.982253 


.017747 


412 


.999980 


27 


2040 


34 


7.995198 


568 


589 


7.995219 


12.004781 


411 


.999979 


26 


2100 


35 


8.007787 


567 


590 


8.007809 


11.992191 


410 


.999977 


25 


2160 


36 


.020021 


567 


591 


.020044 


.979956 


409 


.02 


.999976 


24 


2220 


37 


.031919 


566 


592 


.031945 


.968055 


408 


.02 
.03 


.999975 


23 


2280 


38 


.043501 


566 


593 


.043527 


.956473 


407 


.999973 


22 


2340 


39 


.054781 


566 


593 


.054809 


.945191 


407 


.02 
.02 


.999972 


21 


2400 


40 


.065776 


565 


594 


.065806 


.934194 


406 


.999971 


20 


2460 


41 


8.076500 


565 


595 


8.076531 


11.923469 


405 


.03 

.02 


9.999969 


19 


2520 


42 


.086965 


564 


596 


.086997 


.913003 


404 


.999968 


18 


2580 


43 


.097183 


564 


598 


.097217 


.902783 


402 


.03 
.03 

.02 


.999966 


17 


2640 


44 


.107167 


563 


599 


.107203 


.892797 


401 


.999964 


16 


2700 


45 


.116926 


562 


600 


.116963 


.883037 


400 


.999963 


15 


2760 


46 


.126471 


562 


601 


.126510 


.873490 


399 


,03 
.03 
.02 
.03 
.03 


.999961 


14 


2820 


47 


.135810 


561 


602 


.135851 


.864149 


398 


.999959 


13 


2880 


48 


.144953 


561 


603 


.144996 


.855004 


397 


.999958 


12 


2940 


49 


.153907 


560 


604 


.153952 


.846048 


396 


.999956 


11 


3000 


50 


.162681 


560 


605 


.162727 


.837'273 


395 


.999954 


10 


3060 


51 


8.171280 


559 


607 


8.171328 


11.828672 


393 


.03 
.03 
.03 
.03 
.03 
.03 
.03 
.03 
.03 
.03 


9.999952 


9 


3120 


52 


.179713 


558 


608 


.179763 


.820237 


392 


.999950 


8 


3180 


53 


.187985 


558 


609 


.188036 


.811964 


391 


.999948 


7 


3240 


54 


.196102 


557 


611 


.196156 


.803844 


389 


.999946 


6 


&300 


55 


.204070 


556 


612 


.204126 


.795874 


388 


.999944 


5 


3*60 


56 


.211895 


556 


613 


.211953 


.788047 


387 


.999942 


4 


3120 


57 


.219581 


555 


615 


.219641 


.780359 


385 


.999940 


3 


3480 


58 


.227134 


554 


616 


.227195 


.772805 


384 


.999938 


2 


3540 


59 


.234557 


554 


618 


.234621 


.765379 


382 


.999936 


1 


3600 


60 


8.241855 


553 


619 


8.241921 


11.758079 


381 


9.999934 











4.685 






15.314 








tt 


/ 


Cosine. 


q-l 


Cotang. 


Tang. 


q + l 


Dl" 


Sine. 


/ 



90* 



169 



89* 



LOGABITEMIG SINES, 



178° 



// 


/ 


Sine. 


q-l 


Tang. 


Cotang. 


q + l 


Dl" 


Cosine. 


/ 








4.685 






15.314 








3600 





8.241855 


553! 


619 


8.241921 


11.758079 


381 I 


I 
.03 
.05 
.03 
.03 
.05 
.03 
.03 
.05 
.03 
.05 


9.999934 


60 


3660 


1 


.249033 


552 


620 


.249102 


.750898 


380 


.999932 


59 


3720 


2 


.256094 


551 


622 


.256165 


.743835 


378 


.999929 


58 


3780 


3 


.263042 


551 


623 


.263115 


.736885 


377 


.999927 


57 


3840 


4 


.269881 


550! 


625 


.269956 


.730044 


375 


.999925 


56 


3900 


5 


.276614 


549: 


627 


.276691 


.723309 


373 


.999922 


55 


3960 


6 


.283243 


548 


628 


.283323 


.716677 


372 


.999920 


54 


4020 


7 


.289773 


547 


630 


.289856 


.710144 


370 


.999918 


53 


4080 


8 


.296207 


546 


632 


.296292 


.703708 


368 


.999915 


52 


4140 


9 


.302546 


546 


633 


.302634 


.697366 


367 


.999913 


51 


4200 


10 


.308794 


545 


635 


.308884 


.691116 


365 


.999910 


50 


4260 


11 


8.314954 


544 


637 


8.315046 


11.684954 


363 


.05 
.03 
.05 
.05 
.03 
.05 
.05 
.05 
.05 
.05 


9.999907 


49 


4320 


12 


.321027 


543 ! 


638 


.321122 


.678878 


362 


.999905 


48 


4380 


13 


.327016 


542 


640 


.327114 


.672886 


360 


.999902 


47 


4440 


14 


.332924 


541 ! 


642 


.333025 


.66C975 


358 


.999899 


46 


4500 


15 


.338753 


540 


644 


.338856 


.661144 


356 


.999897 


45 


4560 


16 


.344504 


539 


646 


.344610 


.655390 


354 


.999894 


44 


4620 


17 


.350181 


539 


648 


.350289 


.649711 


352 ' 


.999891 


43 


4680 


18 


.355f83 


538 


649 


.355895 


.644105 


351 


.999888 


42 


4740 


19 


.361315 


537 


651 


.361430 


.638570 


349 


.999885 


41 


4800 


20 


.366777 


536 : 


653 


.366895 


.633105 


347 


.999882 


40 


4860 


21 


8.372171 


535 


655 


8.372292 


11.627708 


345 


.05 
.05 
.05 
.05 
.05 
.05 
.05 
.05 
.07 
.05 


9.999879 


39 


4920 


22 


.377499 


534 


657 


.377622 


.622378 


343 


.999876 


38 


4980 


23 


.382762 


533 


659 


.382889 


.617111 


341 


.£99873 


37 


5040 


24 


.387962 


532 


661 


.388092 


.611908 


339 


.999870 


36 


5100 


25 


.393101 


531 


663 


.393234 


.606766 


337 


.999867 


35 


5160 


26 


.398179 


530 


666 


.398315 


.601685 


334 


.999864 


34 


5220 


27 


.403199 


529 


668 


.403338 


.596C62 


332 


.999861 


33 


5280 


28 


.408161 


527 


670 


.408304 


.591696 


330 


.999858 


• 32 


5340 


29 


.413068 


526 


672 


.413213 


.586787 


328 


.999854 


31 


5400 


30 


.417919 


525 


674 


.418068 


.581932 


326 


.999851 


30 


5460 


31 


8.422717 


524 


676 


8.422869 


11.577131 


324 


.05 
.07 
.05 
.05 
.07 
.05 
.07 
.05 
.07 
.07 


9.999848 


29 


5520 


32 


.427462 


523 


679 


.427618 


.572382 


321 


.999844 


28 


5580 


33 


.432156 


522 


681 


.432315 


.567685 


319 


.999841 


27 


5640 


34 


.436800 


521 


683 


.436962 


.563038 


317 


.999838 


26 


5700 


35 


.441394 


520 


685 


.441560 


.558440 


315 


.999834 


25 


5760 


36 


.445941 


518 


688 


.446110 


.553890 


312 


.999831 


24 


, 5820 


37 


.450440 


517 


690 


.450613 


.549387 


310 


.999827 


23 


5880 


38 


.454893 


516 


693 


.455070 


.544930 


807 


.999824 


22 


5940 


39 


.459301 


515 


695 


.459481 


.540519 


305 


.999820 


21 


6000 


40 


.463665 


514 


697 


.463849 


.536151 


303 


.999816 


20 


6060 


41 


8.467985 


512 


700 


8.468172 


11.531828 


300 


.05 

.07 
.07 
.07 
.07 
.05 
.07 
.07 
.07 
.07 


9.999813 


19 


6120 


42 


.472263 


511 


702 


.472454 


.527546 


298 


.999809 


18 


6180 


43 


.476498 


510 


705 


.476693 


.523307 


295 


.999805 


17 


6240 


44 


.480693 


509 


707 


.480892 


.519108 


293 


.999801 


16 


6300 


45 


.484848 


507 


710 


.485050 


.514950 


290 


.9S9797 


15 


6360 


46 


.488963 


506 


713 


.489170 


.510830 


287 


.999794 


14 


6420 


47 


.493040 


505 


715 


.493250 


.506750 


285 


.999790 


13 


6480 


48 


.497078 


503 


718 


.497293 


.502707 


282 


.999786 


12 


6540 


49 


.501080 


502 


720 


.501298 


.498702 


280 


' .999782 


11 


6600 


50 


.505045 


501 


723 


.505267 


.494733 


277 


.999778 


10 


6660 


51 


8.508974 


499 


726 


8.509200 


11.490800 


274 


.07 
.08 
.07 
.07 
.07 
.07 
.08 
.07 
.07 
.08 


9.999774 


9 


6720 


52 


.512867 


498 


729 


.513098 


.486902 


271 


.999769 


8 


6780 


53 


.516726 


497 


731 


.516961 


.483039 


269 


.999765 


7 


6840 


54 


.520551 


495 


734 


.520790 


.479210 


266 


.999761 


6 


6900 


55 


.524343 


494 


737 


. 524586 


.475414 


263 


.999757 


5 


6960 


56 


.528102 


492 


740 


.528349 


.471651 


260 


.999753 


4 


7020 


57 


.531828 


491 


743 


.532080 


.467920 


257 


.999748 


3 


7080 


58 


.535523 


490 


745 


.535779 


.464221 


255 


.999744 


2 


7140 


59 


.539186 


488 


748 


.539447 


.460553 


252 


.999740 


1 


7200 


60 


8.542819 


487 
4.( 


751 

585 


8.543084 


11.456916 


249 
15.314 


9.999735 





// 


/ 


Cosine. 


Q- 


-I 


Cotang. 


Tang. 


q + l\ 


Dl- 


Sine. 


' ! 



91° 



170 



68* 



COSINES, TANGENTS, AND COTANGENTS. 



177° 





1 

2 

3 
4 
5 
6 

7 

8 

9 

10 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

21 
22 

23 
24 
25 
26 

27 
28 



31 
32 
33 
34 
35 
36 
37 
38 
30 
40 

41 
42 
43 

44 
45 
46 
47 
48 
49 
50 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



Sine. D. 1". \ Cosine. D. 1". j Tang. D. 1'. Cotang, 



8 542819 
.546422 
.549995 
.553539 
.557054 
.560540 
.563999 
.567431 
.570836 
.574214 
.577566 

3.580892 
.584193 

.587469 
.590721 
.593948 
.597152 
.600332 
.603489 
.606623 
.609734 

3.612823 
.615891 
.618937 
.621962 
.624965 
.627948 
.630911 
.6:33854 
.636776 
.639680 

5.642563 
.64.5428 
.648274 
.651102 
.653911 
.656702 
.659475 
.662230 
.664968 
.667689 

t. 670393 
.673080 
.675751 
.678405 
.681043 



.686272 
.688863 
.6914-8 



5.696543 
.699073 
.701589 
.704090 
.706577 
.709049 
.711507 
.713952 
.716383 
!. 718800 



Cosine. D. 1 



60.05 
59.55 
59.07 
58.58 
58.10 
57.65 
57.20 
56.75 
56.30 
55.87 
55.43 

55.02 
54.60 
54.20 
53.78 
53.40 
53.00 
52.62 
52.23 
51.85 
51.48 

51.13 

50.77 
50.42 
50.05 
49.72 
49.38 
49.05 
48.70 
48.40 
48.05 

47.75 
47.43 
47.13 
46.82 
46.52 
46.22 
45.92 
45.63 
45.35 
45.07 

44.78 
44.52 
44.23 
43.97 
43.70 
43.45 
43.18 
42.92 
42.67 
42.42 

42.17 
41.93 
41.68 
41.45 
41.20 
40.97 
40.75 
40.52 
40.28 



9.9997:35 
.999731 
.999726 
.999722 
.999717 
.999713 
.999708 
.999704 
.999699 



9.999685 
.999680 
.999675 
.999670 
.999665 
.999660 
.999655 
.999650 
.999645 
i .999640 

i 9.999635 
j .999629 

.999624 
I .999619 
! .999614 

.999608 
; .999603 
: .999597 

.999592 

.999586 

9.999581 
.999575 
.999570 
.999564 
.999558 
.999553 
.999547 
.999541 
.999535 
.999529 

9.999524 
.999518 
.999512 
.999506 
.999500 
.999493 
.999487 
.999481 
.999475 
.999469 

9.999463 
.909456 
.999450 
.999443 
.999437 
.999431 
.999424 
.999418 
.999411 

9.999404 



10 
OS 
10 
06 

10 

OS 

10 
06 
10 
10 

<!S 

10 
10 

10 
10 

03 

10 
10 
10 
10 
12 
10 
10 
10 
10 
10 

12 
10 
12 
10 
10 
12 
10 
12 
12 



Sine. 



92° 



d. r 



171 



8.5430^4 
.546691 
.550268 
.553817 
.557336 
.560828 
.564291 
.567727 
.571137 
.574520 
.577877 

8.581208 
.5&4514 
.587795 
.591051 
.594283 
.597492 
.600677 
.603839 
.606978 
.610094 

3.613189 
.616262 
.619313 
.622-343 
.625352 
.628340 
.631308 
.6:34256 
.637184 
.640093 

3.642982 
.645853 
.648704 
.651537 

.654352 
.657149 
.659928 
.662689 
.665433 
.668160 

1.670870 
.673563 
.676239 
.678900 
.681544 
.684172 
.686784 
.689381 



.694529 

5.697081 
.699617 
.702139 
.704646 
.707140 
.709618 
.712083 
.7145&4 
.716972 
.719396 



60.12 
59.62 
59.15 
58.65 
58.20 
57.72 
57.27 
56.83 
56.38 
55.95 
55.52 

55.10 
54.68 
54.27 
53.87 
53.48 
53.08 
52.70 
52.32 
51.93 
51.58 

51.22 
50.85 
50.50 
50.15 
49.80 
49.47 
49.13 
48.80 
48.48 
48.15 

47.85 
47.52 
47.22 
46.92 
46.62 
46.32 
46.02 
45.73 
45.45 
45.17 

44.88 
44.60 
44.35 
44.07 
43.80 
43.53 
43.28 
43.03 
42.77 
42.53 

42.27 
42.03 
41.78 
41.57 
41.30 
41.08 
40.85 
40.63 
40.40 



11.456916 60 

.453309 59 

.449732 58 

.446183 57 

.442664 56 

.439172 55 

.435709 54 

.432273 53 

.428863 52 

.4254*0 51 

.422123 50 

11.418792 49 
.415486 I 48 
.412205 47 
.408949 46 
.405717 45 
.402508 44 
.399323 43 
.396161 42 
.393022 41 
40 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 

11.357018 I 29 
..354147 i 28 
.351296 27 
.&48463 ; 26 
.345648 25 
.&42851 24 
.340072 23 
.337311 22 
.334567 21 
.331840 20 



11.386811 
.383738 
.380687 
.377657 
.374648 
.371660 



.365744 
.362816 
.359907 



11.329130 

.326437 
.323761 
.321100 
.318456 
.315828 
.313216 
.310619 
.308037 
.305471 

11.302919 



.297861 
.295354 



.290382 
.287917 
.285466 



11.280604 



Cotang. i D. 1*. I Tang. 



87* 



LOGABITHMIG SINES, 



176 a 



40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



Sine. 



D. 1". Cosine. D. 1". Tang. D. 1". Cotang, 



8.718800 
.721204 
.723595 

.725972 
.728337 



.733027 
.735354 

.737667 



.742259 
1.744536 



.749055 
.751297 
.753528 
.755747 
. 757955 
.760151 
.762337 
.764511 

3. 766675 
.768828 
.770970 
.773101 
.775223 
.777333 
.779434 
.781524 
.783605 
.785675 

3.787736 
.789787 



.795881 
.797894 
.799897 
.801892 
.803876 
.805852 

8.807819 
.809777 
.811726 
.813667 
.815599 
.817522 
.819436 
.821343 
.823240 
.825130 

8.827011 
.828884 
.830749 
.832607 
.834456 
.836297 
.838130 
.839956 
.841774 

8.843585 

Cosine. 



40.07 
39.85 
39.62 
39.42 
39.18 
38.98 
38.78 
38.55 
38.37 
38.17 
37.95 

37.77 
37.55 
37.37 
37.18 
36.98 
36.80 
36.60 
36.43 
36.23 
36.07 

35.88 
35.70 
35.52 
35.37 
35.17 
35.02 
34.83 
34.68 
34.50 
34.35 

34.18 
34.02 
33.85 
33.70 
33.55 
33.38 
33.25 
33.07 
32.93 
. 32.78 

32.63 
32.48 
32.35 
32.20 
32.05 
31.90 
31.78 
31.62 
31.50 
31.35 

31.22 
31.08 
30.97 
30.82 
30.68 
30.55 
30.43 
30.30 
30.18 

D 1". 



9.999404 
.999398 
.999391 
.999384 
.999378 
.999371 
.999364 
.999357 
.999350 
.999343 



9.999329 
.999322 
.999315 
.999308 
.999301 
.999294 
.999287 
.999279 
.999272 
.999265 

9.999257 
.999250 
.999242 
.999235 
.999227 
.999220 
.999212 
.999205 
.999197 
.999189 

9.999181 
.999174 
.999166 
.999158 
.999150 
.999142 
.999134 
.999126 
.999118 
.999110 

9.999102 

.999094 
.999086 
.999077 
.999069 
.999061 
.999053 
.999044 
.999036 
.999027 

9.999019 
.999010 
.999002 
.998993 
.998984 
.998976 
.998967 



9.998941 
Sine. 



.10 
.12 
.12 
.10 
,12 
.12 
.12 
.12 
.12 
.12 
.12 

.12 
.12 
.12 
.12 
.12 
.12 
.13 
.12 
.12 
.13 

.12 
.13 
.12 
.13 
.12 
.13 
.12 
.13 
.13 
.13 

.12 
.13 
.13 
.13 
.13 
.13 
.13 
.13 
.13 
.13 

.13 
.13 
.15 
.13 
.13 
.13 
.15 
.13 
.15 
.13 

.15 
.13 
.15 
.15 
.13 
.15 
.15 
.13 
.15 



93 Q 



D. 1*. 
172 



1.719396 
.721806 
.724204 
.726588 
.728959 
.731317 
.733663 
.735996 
.738317 
.740626 
.742922 

1.745207 
.747479 
.749740 
.751989 
.754227 
.756453 
.758668 
.760872 
.763065 
.765246 

.767417 
.769578 

.771727 



775995 
778114 

780222 



.784408 
.786486 

8.788554 
.790613 
.792662 
.794701 
.796731 
.798752 
.800763 
.802765 
.804758 
.806742 

8.808717 
.810683 
.812641 
.814589 
.816529 
.818461 
.820384 
.822298 
.824205 
.826103 

8.827992 
.829874 
.831748 
.833613 
.835471 
.837321 
.839163 
.840998 
.842825 

8.844644 

Cotang. 



40.17 
39.97 
39.73 
39.52 
39.30 
39.10 
38.88 
38.68 
38.48 
38.27 
38.08 

37.87 
37.68 
37.48 
37.30 
37.10 
36.92 
36.73 
36.55 
36.35 
36.18 

36.02 
35.82 
35.65 
35.48 
35.32 
35.13 
34.97 
34.80 
34.63 
34.47 

34.32 

34.15 
33.98 
33.83 
33.68 
33.52 
33.37 
33.22 
33.07 
32.92 

32.77 
32.63 
32.47 
32.33 
32.20 
32.05 
31.90 
31.78 
31.63 
31.48 

31.37 
31.23 
31.08 
30.97 
30.83 
30.70 
30.58 
30.45 
30.32 

D. 1'. 



11.280604 
.278194 
.275796 
.273412 
.271041 



.266337 
.264004 
.261683 
.259374 
.257078 

11.254793 
.252521 
.250260 
.248011 
.245773 
.243547 
.241332 
.239128 
.236935 
.234754 

11.232583 
.230422 
.228273 
.226134 
.224005 
.221886 
.219778 
.217680 
.215592 
.213514 

11.211446 

.209387 
.207338 
.205299 
.203269 
.201248 
. 199237 
.197235 
.195242 
.193258 

11.191283 
.189317 
.187a39 
.185411 
.183471 
.181539 
.179616 
.177702 
.175795 
.173897 

11.172008 
.170126 
.168252 
.166387 
.164529 
.162679 
.160837 
.159002 
.157175 

11.155356 

Tang. 



86° 



COSINES, TANGENTS, AND COTANGENTS. 



175° 



Sine. D. 1". ! Cosine. D. 1". Tang. D. 1". Cotang. 






8.843585 


1 


.845387 


2 


.&47183 


3 


.818971 


4 


.850751 


5 


.852525 


6 


.854291 


7 


.850049 


8 


.85:801 


9 


.859546 


10 


.861283 


11 


8.863014 


12 


.864738 


13 


.866455 


14 


.868165- 


15 


.869868 


16 


.871565 


17 


.873255 


18 


.8749:38 


19 


.876615 


20 


.878285 


21 


8.879949 


22 


.881607 


23 


.883258 


24 


.884903 


25 


.886542 


26 


.888174 


27 


.889801 


28 


.891421 


29 


.893035 


30 


.894643 


31 


8.896246 


32 


.897812 


33 


.899432 


34 


.901017 


35 


.902596 


36 


.904169 


37 


.905736 


38 


.907297 


39 


.908853 


40 


.910404 


41 


8.911949 


42 


.913488 


43 


.915022 


44 


.916550 


45 


.918073 


46 


.919591 


47 


.921103 


48 


.922610 


49 


.924112 


50 


.925609 


51 


8.927100 


52 




53 


.930068 


54 


.931544 


55 


.9:33015 


56 


.9:34481 


57 


.935942 


58 


.937398 


59 


.938850 


60 


8.940296 



I Cosine. D. 1". 



30.03 
29.93 
29.80 
29.67 
29.57 
29.43 
29.30 
29.20 
29.08 
28.95 
28.85 

28.73 
28.62 
28.50 
28.38 
28.28 
28.17 
28.05 
27.95 
27.83 
27.73 

27.63 
27.52 
27.42 
27.32 
27.20 
27.12 
27.00 
26.90 
26.80 
26.72 

26.60 
26.50 
26.42 
26.32 
26.22 
26.12 
26.02 
25.93 
25.85 
25.75 

25.65 
25.57 
25.47 
25.38 
25.30 
25.20 
25.12 
25.03 
24.95 
24.85 

24.78 
24.68 
24.60 
24.52 
24.43 
24.35 
24.27 
24.20 
24.10 



9.998941 
.998932 
.998923 
.998914 
.998905 



.998887 
.998878 
.998869 
.998860 
.998851 

9.998541 
.998832 
.998823 
.998813 
.998804 
.998795 
.998785 
.998776 
.998766 
.998757 

9.998747 
.998738 
.998728 
.998718 
.998708 
.998699 
.998689 
.998679 
.998669 
.99S659 

9.998649 
.998639 
.998629 
.998619 
.998609 
.998599 
.998589 
.998578 
.998568 
.998558 

9.998548 
.998537 
.998527 
.998516 
.998506 
.998495 
.998485 
.998474 
.998464 
.998453 

9.998442 
.998431 

.998421 
.998410 
.998399 
.998388 
.998377 



.998355 

9.998344 



1.844644 
.846455 
.848260 
.850057 
.851846 
.853628 
.855403 
.857171 



.860686 
.862433 

8.864173 
.865906 
.867632 
.869351 
.871064 
.872770 
.874469 
.876162 
.877849 
.879529 

8.881202 



.884530 
.886185 

.887833 

.889476 ! 

.891112 

.892742 

.894366 

.895984 

.897596 
.899203 
.900803 
.902398 
.903987 
.905570 
.907147 
.908719 
.910285 
.911846 

1.913401 
.914951 
.916495 
.9180:34 
.919568 
.921096 
.922619 
.924136 
.925649 
.927156 

1.928658 
.930155 
.931647 
.9:331:34 
.934616 
.936093 
.937565 
.939032 
.940494 
.941952 



30.18 
30.08 
29.95 
29.82 
29.70 
29.58 
£9.47 
29.35 
29.23 
29 r 12 
29.00 

28.88 
28.77 
28.65 
28.55 
28.43 
28.32 
28.22 
28.12 
28.00 
27.88 

27.78 
27.68 
27.58 
27.47 
27.38 
27.27 
27.17 
27.07 
26.97 
26.87 

26.78 
26.67 
26.58 
26.48 
26.38 
26.28 
26.20 
26.10 
26.02 
25.92 

25.83 
25.73 
25.63 
25.57 
25.47 
25.38 
25.28 
25.22 
25.12 
25.03 

24.95 
24.87 
24.78 
24.70 
24.62 
24.53 
24.45 
24.37 
24.30 



11.155356 
.153545 
.151740 
.149943 
.148154 
.146372 
.144597 
.142829 
.141068 
.139314 
.137567 

11.135827 
.134094 
. 132368 
.130649 
.128936 
.127230 
.125531 
.123838 
.122151 
.120471 

11.118798 

.117131 38 

.115470 37 

.113815 36 

.112167 35 

.110524 34 

.108888 33 

.107258 32 

.105634 31 

.104016 30 

11.102404 29 

.100797 28 

.099197 27 

.097602 26 

.096013 25 



11.086599 
.085049 
.083505 
.081966 
.080432 
.078904 
.077381 
.075864 
.074351 
.072844 

11.071342 
.069845 
.068353 



.0653&4 
.063907 
.062435 

.060968 

.059506 

11.058048 



60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 

49 
48 
'47 
46 
45 
44 
43 
42 
41 
40 



.094430 
.092853 
.091281 
.089715 21 
.088154 20 



94° 



Sine. D. 1". . Cotang. D. 1'. | Tang. 

173 85° 



LOGARITHMIC SINES, 



174° 



9 

10 

11 
12 

13 
14 
15 
16 
17 
18 
19 
20 

21 
22 
23 
24 
25 
23 
27 
28 
29 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



Sine. 



D. 1". 



8.940296 
.941738 
.943174 
.944606 
.946034 
.947456 
.948874 
.950287 
.951696 
.953100 
.954499 

8.955894 
.957284 
.958670 
.960052 
.961429 
.962801 
.964170 
.965534 
.966893 
.968249 

8.969600 
.970947 
.972289 
.973628 
.974962 
.976293 
.977619 
.978941 



.981573 

8.982883 
.984189 
.985491 
.986789 



.989374 
.990660 
.991943 
.993222 
.994497 

8.995768 
.997036 
.998299 
8.999560 
9.000816 
.002069 
.003318 
.004563 
.005805 
.007044 

9.008278 
.009510 
.010737 
.011962 
.013182 
.014400 
.015613 
.016824 
.018031 

9.019235 

Cosine. 



I 
24.03 
23.93 | 

23.87 
23.80 
23.70 
23.63 
23.55 
23.48 
23.40 
23.32 
23.25 

23.17 
23.10 
23.03 
22.95 
22.87 
22.82 
22.73 
22.65 
22.60 
22.52 

22.45 
22.37 
22.32 
22.23 
22.18 
22.10 
22.03 
21.97 
21.90 
21.83 

21.77 
21.72 
21.63 
21.57 
21.52 
21.43 
21.38 
21.32 
21.25 
21.18 

21.13 
21.05 
21.02 
20.93 
20.88 
20.82 
20.75 
20.70 
20.65 
20.57 

20.53 
20.45 
20.42 
20.33 
20.30 
20.22 
20.18 
20.12 
20.07 

D. 1'. 



Cosine. 



D. 1\ 



9.99S344 
.998333 
.998322 
.998311 
.998300 
.998289 
.998277 



.998255 
.998243 



.998209 
.998197 
.998186 
.998174 
.998163 
.998151 
.998139 
.998128 
.998116 

9.998104 



.998044 
.998032 
.998020 
.998008 
.997996 

9.997984 
.997972 
.997959 
.997947 
.997935 
.997922 
.997910 
.997897 
.997885 
.997872 

9.997860 
.997847 
.997835 
.997822 
.997809 
.997797 
.997784 
.997771 
.997758 
.997745 

9.997732 
.997719 
.997706 
.997693 
.997680 
.997667 
.997654 
.997641 
.99^628 

9.997614 

Sine. 



95° 



.18 
.18 
.18 
.18 
.18 
.20 
.18 
.18 
.20 
.18 
.20 

.13 
.20 
.18 
.20 
.18 
.20 
.20 
.18 
.20 
.20 

.20 
.20 
.20 
.20 
.20 
.20 
.20 
.20 
.20 
.20 

.20 
.22 
.20 
.20 
.22 
.20 
.22 
.20 
.22 
.20 

.22 
.20 
.22 
.22 
.20 
.22 
.22 
.22 
.22 
.22 

.22 

.22 
.22 
.22 
.22 
.22 
.22 
.22 
.23 

D. 1". 

174 



Tang. 



8.941952 
.943404 
.944852 
.946295 
.947734 
.949168 
.950597 
.952021 
.953441 
.954856 
.956267 

8.957674 
.959075 
.960473 
.961866 
.963255 
.964639 
.966019 
.967394 
.968766 
.970133 

8.971496 
.972855 
.974209 
.975560 
.976906 
.978248 
.979586 
.980921 
.982251 
.983577 



.986217 
.987532 
.988842 
.990149 
.991451 
.992750 
.994045 
.995337 
.996624 

8.997908 
8.999188 
9.000465 
.001738 
.003007 
.004272 
.005534 
.006792 
.008047 
.009298 

9.010546 
.011790 
.013031 
.014268 
.015502 
.016732 
.017959 
.019183 
.020403 

9.021620 

Cotang. 



d. r. 



24.20 
24.13 
24.05 
23.98 
23.90 
23.82 
23.73 
23.67 
23.58 
23.52 
23.45 

23.35 
23.30 
23.22 
23.15 
23.07 
23.00 
22.92 
22.87 
22.78 
22.72 

22.65 
22.57 
22.52 
22.43 
22.37 
22.30 
22.25 
22.17 
22.10 
22.03 

21.97 
21.92 
21.83 
21.78 
21.70 
21.65 
21.58 
21.53 
21.45 
21.40 

21.33 
21.28 
21.22 
21.15 
21.08 
21.03 
20.97 
20.92 
20.85 
20.80 

20.73 
20.68 
20.62 
20.57 
20.50 
20.45 
20.40 
20.33 
20.28 

d. r. 



Cotang. 



11.058048 
.056596 
.055148 
.053705 
.052266 
.050832 
.049403 
.047979 
.046559 
.045144 
.043733 

11.042326 
.040925 
.039527 
.038134 
.036745 
.035361 
.033981 
.032606 
.0312:34 
.029867 

11.C28504 
.027145 
.025791 
.024440 
.023094 
.021752 
.020414 
.019079 
.017749 
.016423 

11.015101 

.013783 
.012468 
.011158 
.009851 
.008549 
.007250 
.005955 
.004663 
.003376 

11.C02092 
11.000812 
10.999535 



.995728 
.944466 
.993208 
.991953 
.990702 

10.989454 
.988210 
.986969 
.985732 
.984498 
.983268 
.982041 
.980817 
.979597 

10.978380 

Tang. 



84° 



6° 



COSINES, TANGENTS, AND COTANGENTS. mo 



' Sine. D. 1*. I Cosine. D. 1\ Tang. D. 1*. Cotang. ' 



31 



40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



9.019235 
.020435 
.021632 
.022825 
.024016 
.025203 
.026386 
.027567 
.028744 
.029918 
.031089 

9.032257 
.033421 

.034582 
.035741 



.038048 
.039197 
.040342 
.041485 
.042625 

9.043762 
.044895 
.046026 
.047154 
.048279 
.049400 
.050519 
.051635 
.052749 
.053859 

9.054966 
.056071 
.057172 
.058271 
.059367 
.060460 
.061551 



.063724 
.064806 

9.065885 
.066962 
.068036 
.069107 
.070176 
.071242 
.072306 
.073366 
.074424 
.075480 

9.076533 
.077583 
.078631 
.079676 
.080719 
.081759 
.082797 
.083832 
.084864 

9.085894 



20.00 
19.95 
19.88 
19.85 
19.78 
19.72 
19.68 
19.62 
19.57 
19 52 
19.47 

19.40 i 
19.35 
19.32 
19.25 

19.20 
19.15 
19.08 
19.05 
19.00 
18.95 

18.88 
18.85 
18.80 
18.75 
18.68 
18.65 
18.60 
18.57 
18.50 
18.45 

18.42 
18.35 
18.32 
18.27 
18.22 
18.18 
18.13 
18.08 
18.03 
17.98 

17.95 
17.90 
17.85 
17.82 
17.77 
17.73 
17.67 
17.63 
17.60 
17.55 

17.50 
17.47 
17.42 
17.38 
17.33 
17.30 
17.25 
17.20 
17.17 



' Cosine. | D. I s , 



9.997014 
.997601 
.997588 
.997574 
.997561 
.997547 
.997534 
.997520 
.997507 
.997493 
.997480 

9.997466 
.997452 
.997439 
.997425 
.997411 
.997397 
.997383 
.997369 
.997355 
.997341 

9.997327 
.967313 
.997299 
.997285 
.997271 
.997257 
.997242 
.997228 
.997214 
.997199 

9.997185 
.997170 
.997156 
.997141 
.997127 
.997112 
.997098 
.997083 
.997068 
.997053 

9.997039 
.997024 
.997009 
.996994 
.996979 
.996964 



.996934 
.996919 
.996904 



.996874 



.996843 



.996812 
.996797 
.996782 
.996766 
9.996751 



.22 
.22 
.23 
.22 
.23 
.22 
.23 
.22 
.23 
.22 
.23 

.23 
.22 
.23 
.23 
.23 
.23 
.23 
.23 
.23 
.23 

.23 
.23 
.23 
.23 
.23 
.25 
.23 
.23 
.25 
.23 

.25 
.23 
.25 
.23 
.25 
.23 
.25 
.25 
.25 
.23 

.25 
.25 
.25 
.25 
.25 
.25 
.25 
.25 
.25 
.25 

.25 
.27 
.25 

.27 
.27 
.25 
.25 
.27 
.25 



Sine. 



96° 



D. 1", 



175 



9.021620 
.022834 
.024044 
.025251 
.026455 
.027655 
.028852 
.030046 
.031237 
.032425 
.033609 

9.034791 
.035969 
.037144 
.038316 
.039485 
.040651 
.041813 
.042973 
.044130 
.045284 

1.046434 
.047582 
.048727 
.049869 
.051008 
.052144 
.053277 
.054407 
.055535 
.056659 

1.057781 
.058900 
.060016 
.061130 
.062240 
.063348 
.064453 
.065556 
.066655 
.067752 



.071027 
.072113 
.073197 
.074278 
.075356 
.076432 
.077505 
.078576 

9.079644 
.080710 
.081773 



.083891 
.084947 
.086000 
.087050 



9.089144 



20.23 
20.17 
20.12 
20.07 
20.00 
19.95 
19.90 
19.85 
19.80 
19.73 
19.70 

19.63 
19.58 
19.53 
19.48 
19.43 
19.37 
19.33 
19.28 
19.23 
19.17 

19.13 
19.08 
19.03 
18.98 
18.93 
18.88 
18 83 
18.80 
18.73 
18.70 

18.65 
18.60 
18.57 
18.50 
18.47 
18.42 
18.38 
18.32 
18.28 
18.25 

18.20 
18.15 
18.10 
18.07 
18.02 
17.97 
17.93 
17.88 
17.85 
17.80 

17.77 
17.72 
17.67 
17.63 
17.60 
17.55 
17.50 
17.47 
17.43 



10.978380 
.977166 
.9759.56 
.974749 
.973545 
.972345 
.971148 
.969954 
.968763 
.967575 
.966391 

10.965209 
.964031 



.961684 
.960515 
.959349 
.958187 
.957027 
.955870 
.954716 

10.953566 
.952418 
.951273 
.950131 
.948992 
.947856 
.946723 
.945593 
.944465 
.943341 

10.942219 
.941100 
.939984 
.938870 
.937760 
.936652 
.935547 
.934444 
.933345 



10.931154 
.930062 
.928973 

.927887 



.925722 
.924644 



.922495 
.921424 

10.920356 
.919290 
.918227 
.917167 
.916109 
.915053 
.914000 
.912950 
.911902 

10.910856 



Cotang. | D. 1". | Tang. 



83° 



LOGARITHMIC SINES, 



172° 



' Sine. D. 1'. Cosine. D. 1". Tang. D. 1". Cotang. 





1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



97° 



9.085894 
.086922 

.087947 
.088970 



.091008 
.092024 



.094047 
.095056 
.096062 

9.097065 



.099065 
.100062 
.101056 
.102048 
.103037 
.104025 
.105010 
.105992 

9.106973 
.107951 
.108927 
.109901 
.110873 
.111842 
.112809 
.113774 
.114737 
.115698 

9.116656 
.117613 
.118567 
.119519 
.120469 
.121417 
.122362 
.123306 
.124248 
.125187 

9.126125 
.127060 
.127993 
.128925 
.129854 
.130781 
.131706 
.132630 
.133551 
.134470 

9.135387 
.136303 
.137216 
.138128 
.139037 
.139944 
.140850 
.141754 
.142655 

9.143555 

Cosine. 



17.13 


17.08 


17.05 


17.00 


16.97 


16.93 


16.88 


16.83 


16.82 


16.77 


16.72 


16.68 


16.65 


16.62 


16.57 


16.53 


16.48 


16.47 


16.42 


16.37 


16.35 


16.30 


16.27 


16.23 


16.20 


16.15 


16.12 


16.08 


16.05 


16.02 


15.97 


15.95 


15.90 


15.87 


15.83 


15.80 


15.75 


15.73 


15.70 


15.65 


15.63 


15.58 


15.55 


15.53 


15.48 


15.45 


15.42 


15.40 


15.35 


15.32 


15.28 


15.27 


15.22 


15.20 


15.15 


15.12 


15.10 


15.07 


15.02 


15.00 



9.996751 
.996735 
.996720 
.996704 



.996673 
.996657 
.996641 
.996625 
.996610 
.996594 

9.996578 
.996562 
.996546 
.996530 
.996514 



.996465 



.996433 

9.996417 
.996400 
.996384 



.996351 
.996335 
.996318 
.996302 
.996285 



9.996252 
.996235 
.996219 
.996202 
.996185 
.996168 
.996151 
.996134 
.996117 
.996100 

9.996083 
.996066 
.996049 
.986032 
.996015 



.995980 
.995963 
.995946 



9.995911 



.995876 
.995859 
.995811 
.995823 



.995788 

.995771 

9.995753 



D. 1". I Sine. 



.27 
.25 

.27 
.27 
.25 
.27 
.27 
.27 
.25 
.27 
.27 

.27 
.27 
.27 
.27 
.27 
.27 
.28 
.27 
.27 
.27 

.28 
.27 
.27 
.28 
.27 
.28 
.27 
.28 
.27 
.28 

.28 
.27 

.28 
.28 
.28 
.28 
.28 
.28 
.28 
.28 

.28 
.28 
.28 
.28 
.28 
.30 
.28 



.28 

.28 
.30 
.28 
.30 
.30 
.28 
.30 
.28 
.30 



d. r. 



176 



9.089144 
.090187 
.091228 
.092266 
.093302 
.094336 
.095367 
.096395 
.097422 
.098446 



9.100487 
.101504 
.102519 
.103532 
.104542 
.105550 
.106556 
.107559 
.108560 
.109559 

9.110556 
.111551 
.112543 
.113533 
.114521 
.115507 
.116491 
.117472 
.118452 
.119429 

9.120404 
.121377 
.122348 
.123317 
.124284 
.125249 
.126211 
.127172 
.128130 
.129087 

9.130041 
.130994 
.131944 
.132893 
.133839 
.134784 
.135726 
.136667 
.137605 
.138542 

9.139476 
.140409 
.141340 
.142269 
.143196 
.144121 
.145044 
.145966 
.146885 

9.147803 

Cotang. 



17.38 
17.35 
17.30 
17.27 
17.23 
17.18 
17.13 
17.12 
17.07 
17.03 
16.98 

16.95 
16.92 
16.88 
16.83 
16.80 
16.77 
16.72 
16.68 
16.65 
16.62 

16.58 
16.53 
16.50 
16.47 
16.43 
16.40 
16.35 
16.33 
16.28 
16.25 

16.22 
16.18 
16.15 
16.12 
16.08 
16.03 
16.02 
15.97 
15.95 
15.90 

15.88 
15.83 
15.82 
15.77 
15.75 
15.70 
15.68 
15.63 
15.62 
15.57 

15.55 
15.52 
15.48 
15.45 
15.42 
15.38 
15.37 
15.32 
. 15.30 

D. 1". 



10.910856 
.909813 
.908772 
.907734 
.906698 
.905664 
.904633 
.903605 
.902578 
.901554 
.900532 

10.899513 
.898496 
.897481 
.896468 
.895458 
.894450 
.893444 
.892441 
.891440 
.890441 

10.889444 

.888449 
.887457 
• .886467 
.885479 
.884493 
.883509 
.882528 
.881548 
.880571 

10.879596 

.878623 
.877652 
.876683 
.875716 
.874751 
.873789 
.872828 
.871870 
.870913 

10.869959 
.869006 
.868056 
.867107 
.866161 
.865216 
.864274 
.863333 
.862395 
.861458 

10.860524 
.859591 
.858660 
.857731 
.856804 
.855879 
.854956 
.854034 
.853115 

10.852197 

Tang. 



82° 



COSINES,. TANGENTS AND COTANGENTS. 



171° 



o 
1 

2 
3 
4 
5 
6 
7 
8 
9 
10 

11 
12 
13 

14 
15 
16 
17 
18 
19 
20 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

41 
42 

43 
44 
45 
46 
47 
48 
49 
50 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



Sine. 



9.143555 
.144453 
.145349 
.146243 
.147136 
.148026 
.148915 
.149802 
.150686 
.151569 
.152451 

9.153330 
.154208 
.155083 
.155957 
.156830 
.157700 
.158569 
.159435 
.160301 
.161164 

9.162025 

.162885 
. 163743 
.164600 
.165454 
.166307 
.167159 
.168008 
.168856 
.169702 

9.170547 

.171389 
.172230 
.173070 
.173908 
.174744 
.175578 
.176411 
.177242 
.178072 

9.178900 
.179726 
.180551 
.181374 
.182196 
.183016 
.183834 
.184651 
.185466 
.186280 

9.187092 
.187903 
.188712 
.189519 
.190325 
.191130 
.191933 
.192734 
.193534 

9.194332 



D. 1". 



Cosine. 



14.97 
14.93 
14.90 
14.88 
14.83 
14.82 
14.78 
14.73 
14.72 
14.70 
14.65 

14.63 
14.58 
14.57 
14.55 
14.50 
14.48 
14.43 
14.43 
14.38 
14.35 

14.33 
14.30 
14.28 
14.23 
14.2;} 
14.20 
14.15 
14.13 
14.10 
14.08 

14.03 
14.02 
14.00 
13.97 
13.93 
13.90 
13.88 
13.85 
13.83 
13.80 



13.77 
13.75 
13.72 
13.70 
13.67 
13.63 
13.62 
13.58 
13.57 
13.53 

13.52 
13.48 
13.45 
13.43 
13.42 
13.38 
13.35 
13.33 
13.30 



I Cosine, i D. 1". 



9.995753 
.995735 
.995717 
.995699 
.995681 
.995664 
.995646 
.995028 
.995610 
.995591 
.995573 

9.995555 
.995537 
.995519 
.995501 
.995482 
.995464 
.995446 
.995427 
.995409 



9.995372 
.995353 
.995334 

.995316 
.995297 
.995278 
.995260 
.995241 
.995222 
.995203 

9.995184 
.995165 
. 995146 
.995127 
.995108 
.995089 
.995070 
.995051 
.995032 
.995013 

9.994993 
.994974 
.994955 
.994935 
.994916 
.99481*6 
.994877 
.994857 
.994838 
.994818 

9.994798 
.994779 
.994759 
.994739 
.994720 
.994700 



.994660 

.994640 

9.994620 



D. 1'. 



.30 
.30 
.30 
.28 
.30 
.30 
.30 
.32 
.30 
.SO 

.30 
.30 
.30 
.32 
.30 
.30 
.32 
.30 
.32 
.30 

.32 
.32 
.30 
.32 
.32 
.30 
.32 
.32 
.32 
.32 

.32 
.32 
.32 
.32 
.32 
.32 
.32 
.32 
.32 
.33 

.32 
.32 

.33 
.32 
.33 
.32 
.33 
.32 
.33 
.33 



.33 
.33 
.32 
.33 
.33 
.33 
.33 
.33 



Sine. 



£8° 



D. 1\ 



177 



Tang. 



9.147803 
.148718 
.149632 
.150544 
.151454 
.152363 
.153269 
.154174 
.155077 
.155978 
.156877 

9.157775 
.158671 
. 159565 
.160457 
.161347 
.162236 
.163123 
.164008 
.164892 
.165774 

9.166654 
.167532 
.168409 
.169284 
.170157 
.171029 
.171899 
.172767 
.173634 
.174499 

9.175362 
.176224 
.177084 
.177942 
.178799 
.179655 
.180508 
.181360 
.182211 
.183059 

9.183907 

.184752 
.185597 
.186439 
.187280 
.188120 
J 88958 
.189794 
.190629 
.191462 

9.192294 
.193124 
.193953 
.194780 
.195606 
.196430 
.197253 
.198074 
: 198894 

9.199713 



D. 1". 



15.25 
15.23 
15.20 
15.17 
15.15 
15.10 
15.08 
15.05 
15.02 
14.98 
14.97 

14.93 
14.90 
14.87 
14.83 
14.82 
14.78 
14.75 
14.73 
14.70 
14.67 

14.63 
14.62 
14.58 
14.55 
14.53 
14.50 
14.47 
14.45 
14.42 
14.38 

14.37 
14.33 
14.30 
14.28 
14 27 
14.22 
14.20 
14.18 
14.13 
14.13 

14.08 
14.08 
14.03 
14.02 
14.00 
13.97 
13.93 
13.92 
13.88 
13.87 

13.83 
13.82 
13.78 
13.77 
13.73 
13.72 
13.68 
13.67 
13.65 




Cotang. 


' 


10.852197 


60 


.851282 


59 


.850368 


58 


.849456 


57 


.848546 


56 


.847637 


55 


.846731 


54 


.845826 


53 


.844923 


52 


.844022 


51 


.843123 


50 


10.842225 


49 


.841329 


48 


.840435 


47 


.839543 


46 


.838653 


45 


.837764 


44 


.836877 


43 


.835992 


42 


.835108 


41 


.834226 


40 


10.833346 


39 


.832468 


38 


.831591 


37 


.830716 


36 


.829843 


35 


.828971 


34 


.828101 


33 


.827233 


32 


.826366 


31 


.825501 


30 


10.824638 


29 


.823776 


28 


.822916 


27 


.822058 


26 


.821201 


25 


.820345 


24 


.819492 


23 


.818640 


22 


.817789 


21 


.816941 


20 


10.816093 


19 


.815248 


18 


.814403 


17 


.813561 


16 


.812720 


15 


.811880 


14 


.811042 


13 


.810206 


12 


.809371 


11 


.808538 


10 


10.807706 


9 


.806876 


8 


.806047 


7 


.805220 


6 


.804394 


5 


.803570 


4 


.802747 


3 


.801926 


2 


.801106 


1 


10.800287 






Cotang. [ D. 1". | Tang. 



81° 



LOGARITHMIC SINES. 



170° 



Sine. 






9.194332 


1 


.195129 


2 


.195925 


3 


.196719 


4 


.197511 


5 


.198302 


6 


.199091 


7 


.199879 


8 


.200666 


9 


.201451 


10 


.202234 


11 


9.203017 


12 


.203797 


13 


.204577 


14 


.205354 


15 


.206131 


16 


.206906 


17 


.207679 


18 


.208452 


10 


.209222 


20 


.209992 


21 


9.210760 


22 


.211526 


28 


.212291 


24 


.213055 


25 


.213818 


26 


.214579 


27 


.215338 


28 


.216097 


29 


.216854 


30 


.217609 


31 


9.218303 


32 


.219116 


33 


.219868 


34 


.220618 


35 


.221367 


36 


.222115 


37 


.222861 


38 


.223606 


39 


.224349 


40 


.225092 


41 


9.225833 


42 


.226573 


43 


.227311 


44 


.228048 


45 


.228784 


46 


.229518 


47 


.230252 


48 


.230984 


49 


.231715 


50 


.232444 


51 


9.233172 


52 


.233899 


53 


.234625 


54 


.235349 


55 


.236073 


56 


.236795 


57 


.237515 


58 


.238235 


59 


.238953 


60 


9.239670 



Cosine. 



D. 1". 



13.28 
13.27 
13.23 
13.20 
13.18 
13.15 
13.13 
13.12 
13.08 
13.05 
13.05 

13.00 
13.00 
12.95 
12.95 
12.92 
12.88 
12.88 
12.83 
12.83 
12.80 

12.77 
12.75 
12.73 
12.72 
12.68 
12.65 
12.65 
12.62 
12.58 
12.57 

12.55 
12.53 
12.50 
12.48 
12.47 
12.43 
12.42 
12.38 
12.38 
12.35 

12.33 
12.30 
12.28 
12.27 
12.23 
12.23 
12.20 
12.18 
12.15 
12.13 

12.12 
12.10 
12.07 
12.07 
12.03 
12.00 
12.00 
11.97 
11.95 

D. r7 



Cosine. 



9.994620 
.994600 
.994560 
.994560 
.994540 
.994519 
.994499 
.994479 
.994459 
.994438 
.994418 

9.994398 
.994377 
.994357 
.994336 
.994316 
.994295 
.994274 
.994254 
.994233 
.994212 

9.994191 
.994171 
.994150 
.994129 
.994108 
.994087 
.994066 
.994045 
.994024 
.994003 

9.993982 
.993960 
.993939 
.993918 
.993897 
.993875 
.993854 
.993832 
.993811 
.993789 

9.993768 
.993746 
.993725 
.993703 
.993681 
.993660 
.993638 
.993616 
.993594 
.993572 

9.993550 



.993506 
.993484 
.993462 
.993440 
.993418 
.993396 
.993374 
9.993351 

Sine. 



D. 1". 



.33 
.33 
.33 
.33 
.35 
.33 
.33 
.33 
.35 
.33 
.S3 

.35 
.33 
.35 
.33 
.35 
.35 
.33 
.35 
.35 
.35 

.33 
.35 
.35 
.35 
.35 
.35 
.35 
.35 
.35 
.35 

.37 
.35 
.35 
.35 
.37 
.35 
.37 
°5 
!S7 
.35 

.37 
.35 
.37 
.37 
.35 
.37 
.37 
.37 
.37 
.37 

.37 
.37 
.37 
.37 
.37 
.37 
.37 
.37 



99° 



D. 1". 
178 



Tang. 



9.199713 
.200529 
.201345 
.202159 
.202971 
.203782 
.204592 
.205400 
.206207 
.207013 
.207817 

9.208619 
.209420 
.210220 
.211018 
.211815 
.212611 
.213405 
.214198 
.214989 
.215780 

9.216568 
.217356 
.218142 
.218926 
.219710 
.220492 
.221272 
.222052 
.222830 
.223607 

9.224382 
.225156 
.225929 
.226700 
.227471 
.228239 
.229007 
.229773 
.230539 
.231302 

9.232065 



.234345 
.235103 
.235859 
.236614 
.237368 
.238120 
.238872 

9.239622 
.240371 
.241118 
.241865 
.242610 
.243354 
.244097 
.244839 
.245579 

9.246319 

Cotang. 



D. 1". 



13. CO 
13.60 
13.57 
13.53 
13.52 
13.50 
13.47 
13.45 
13.43 
13.40 
13.37 

13.35 
13.33 
13.30 
13.28 
13.27 
13.23 
13.22 
13.18 
13.18 
13.13 

13.13 
13.10 
13.07 
13.07 
13.03 
13.00 
13.00 
12.97 
12.95 
12.92 

12.90 
12.88 
12.85 
12.85 
12.80 
12.80 
12.77 
12.77 
12.72 
12.72 

12.68 
12.67 
12.65 
12.63 
12.60 
12.58 
12.57 
12.53 
12.53 
12.50 

12.48 
12.45 
12.45 
12.42 
12.40 
12.38 
12.37 
12.33 
12.33 

D. 1". 



Cotang. 



10.800287 
.799471 
.798655 
.797841 
.797029 
.796218 
.795408 
.794600 
. 793793 
.792987 
.792183 

10.791381 
.790580 
.789780 



.788185 
.787389 
.786595 



.785011 
.784220 

10.783432 

.782644 
.781858 
.781074 
.780290 
.779508 
.778728 
.777948 
.777170 
.776393 

10.775618 
.774844 
.774071 
.773300 

.772529 
.771761 
.770993 
.770227 
.709461 



10.767935 
.767174 
.766414 
.765655 
.764897 
.764141 



.762632 
.761880 
.761128 

10.760378 
.759629 
.758882 
.758135 
.757390 
.756646 
.755903 
.755161 
.754421 

10.753681 

Tang. 



80° 



1Qo COSINES, TANGENTS, AND COTANGENTS. 169 „ 



Sine. 






9.239G70 


1 


.240383 


2 


.241101 


3 


.241814 


4 


.242526 


5 


.243237 


6 


.243947 


7 


.244656 


8 


.245363 


9 


.246069 


10 


.246775 


11 


9.247478 


12 


.248181 


13 


.248883 


14 


.249583 


15 


.250282 


16 


.250980 


17 


.251677 


18 


.252373 


19 


.253067 


20 


.253761 


21 


9.254453 


22 


.255144 


23 


.2S5&34 


24 


.256523 


25 


.257211 


26 


.257898 


27 


.258583 


28 


.259268 


29 


.259951 


30 


.260633 


31 


9.261314 


32 


.261994 


33 


.262673 


34 


.263351 


,35 


.264027 


36 


.264703 


87 


.265377 


38 


.266051 


39 


.266723 


40 


.267395 


41 


9.268065 


42 


.268734 


43 


.269402 


44 


.270069 


45 


.270735 


46 


.271400 


47 


.272064 


48 


.272726 


49 


.273388 


50 


.274049 


51 


9.274708 


52 


.275367 


53 


.276025 


54 


.276681 


55 


.277337 


56 


.277991 


57 


.278645 


58 


.279297 


59 


.279948 


60 


9.280599 



Cosine. 



D. 1", 



11.93 
11.92 
11.88 
11.87 
11.85 
11.83 
11.82 
11.78 
11.77 
11.77 
11.72 

11.72 
11.70 
11.67 
11.65 
11.63 
11.62 
11.60 
11.57 
11.57 
11.53 

11.52 
11.50 
11.48 
11.47 
11.45 
11.42 
11.42 
11.38 
11.37 
11.35 

11.33 
11.32 
11.30 
11.27 
11.27 
11.23 
11.23 
11.20 
11.20 
11.17 

11.15 
11.13 
11.12 
11.10 
11.08 
11.07 
11.03 
11.03 
11.02 
10-98 

10.98 
10.97 
10.93 
10.93 
10.90 
10.90 
10.87 
10.85 
10.85 



Cosine. 



'.993351 
.993329 
.993307 
.993284 
.993262 
.993240 
.993217 
.993195 
.993172 
.993149 
.993127 

L993104 
.993081 



.993013 
.992990 
.992967 
.992944 
.992921 



.992875 
.992852 



.992783 
.992759 
.992736 
.992713 
.992690 



9.992643 
.992619 
.992596 
.992572 
.992549 
.992525 
.992501 
.992478 
.992454 
.992430 

9.992406 



.992359 



.992311 
.992287 



.992239 
.992214 
.992190 

.992166 
.992142 
.992118 
.992093 



D. 1". 



.992044 
.992020 
.991996 
.991971 
(.991947 



d. r, 



.37 
.37 
.38 
.37 
.37 
.38 
.37 



.37 



.37 
.38 
.38 
.38 
.38 
.38 
.38 
.38 
.38 



.40 



.38 
.40 
.38 

.40 
.38 
.40 
.38 
.40 
.40 
.38 
.40 
.40 
.40 

40 
.38 
.40 
.40 
.40 
.40 
.40 
.42 
.40 
.40 

.40 
.40 
.42 
.40 
.42 
.40 
.40 
.42 
.40 



Tang. D. 1". Cotang. ' 



9.246319 
.247057 
.247794 
.248530 
.249264 
.249998 
.250730 
.251461 
.252191 
.252920 



9.254374 

.255100 
.255824 
.256547 
.257269 
.257990 
.258710 
.259429 
.260146 
.260863 

9.261578 
.262292 
.263005 
.263717 
.264428 
.265138 
.265847 
.266555 
.267261 
.267967 

9.268671 
.269375 
.270077 
.270779 
.271479 
.272178 
.272876 
.273573 
.274269 
.274964 

9.275658 
.276351 
.277043 
.277734 
.278424 
.279113 
.279801 



.281174 
.281858 

9.282542 
.283225 
.283907 
.2&1588 
.285268 
.285947 
.286624 
.287301 
.287977 

9.288652 



12.30 
12.28 
12.27 
12.23 
12.23 
12.20 
12.18 
12.17 
12.15 
12.13 
12.10 

12.10 
12.07 
12.05 
12.03 
12.02 
12.00 
11.98 
11.95 
11.95 
11.92 

11.90 
11.88 
11.87 
11.85 
11.83 
11.82 
11.80 
11.77 
11.77 
11.73 

11.73 
11.70 
11.70 
11.67 
11.65 
11.63 
11.62 
11.60 
11.58 
11.57 

11.55 
11.53 
11.52 
11.50 
11.48 
11.47 
11.45 
11.43 
11.40 
11.40 

11.38 
11.37 
11.35 
11.33 
11.32 
11.28 
11.28 
11.27 
11.25 



100° 



Sine. D. 1". I Cotang. I D. 1\ 



10.753681 
.752943 
.752206 
.751470 
.750736 
.750002 
.749270 
.748539 
.747809 
.747080 
.746352 



10.745626 49 

.744900 48 

.744176 47 

.743453 46 

.742731 45 
.742010 ! 44 

.741290 43 

.740571 42 
.739854 I 41 

.739137 40 

10.738422 39 

.737708 38 

.736995 37 

.736283 36 

.735572 35 

.734862 34 
.734153 I 33 

.733445 32 

.732739 31 

.732033 30 

10.731329 29 

.730625 28 
.729923 i 27 

.729221 26 

.728521 25 

.727822 24 

.727124 23 
.726427 
.725731 
.725036 

10.724342 19 

.723649 18 
.722957 | 17 

.722266 16 

.721576 15 

.720887 14 
.720199 
.719512 
.718826 
.718142 



10.717458 
.716775 
.716093 
.715412 
.714732 
.714053 
.713376 
.712699 
.712023 

10.711348 



Tang. 



79° 



11° 



LOGARITHMIC SINES, 



168° 



10 

n 

12 
13 
14 
15 
16 
17 
18 
19 
20 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

51 
52 
53 
54 
65 
56 
57 
58 
59 



Sine. 



.281248 
.281897 
.282544 
.283190 



.284480 
.285124 
.285766 



.287048 

9.287688 
.288326 
.288964 
.289600 
.290236 
.290870 
.291504 
.292137 
.292768 
.293399 

9.294029 
.294658 
.295286 
.295913 
.296539 
.297164 
.297788 
.298412 
.299034 
.299655 

9.300276 



d. i\ 



.301514 
.302132 
.302748 
.303364 
.303979 
.304593 
.305207 



9.306430 
.307041 
.307650 



.309474 
.310080 
.310685 
.311289 
.311893 

9.312495 

.313097 



.314297 
.314897 
.315495 
.316092 



.317284 
9.317879 

Cosine. 



10.82 
10.82 
10.78 
10.77 
10.77 
10.73 
10.73 
10.70 
10.70 
10.67 
10.67 

10.63 
10.63 
10.60 
10.60 
10.57 
10.57 
10.55 
10.52 
10.52 
10.50 

10.48 
10.47 
10.45 
10.43 
10.42 
10.40 
10.40 
10.37 
10.35 
10.35 

10.32 
10.32 
10.30 
10.27 
10.27 
10.25 
10.23 
10.23 
10.20 
10.18 

10.18 
10.15 
10.15 
10.13 
10.12 
10.10 
10.08 
10.07 
10.07 
10.03 

10.03 
10.02 
9.98 
10.00 
9.97 
9.95 
9.95 
9.92 
9.92 

D. r. 



Cosine. 



9.991947 
.991922 
.991897 
.991873 
.991848 
.991823 
.991799 
.991774 
.991749 
.991724 



9.991674 
.991649 
.991624 
.991599 
.991574 
.991549 
.991524 



.991473 
.991448 

9.991422 
.991397 
.991372 
.991346 
.991321 
.991295 
.991270 
.991244 
.991218 
.991193 

9.991167 
.991141 
.991115 
.991090 
.991064 



.991012 



.990960 
.990934 



.990855 



.990777 
.990750 
.990724 
.990697 
.990671 

). 990645 
.990618 
.990591 
.990565 
.990538 
.990511 
.990485 
.990458 
.990431 

). 990404 

Sine. 



D. i\ 



101° 



.42 
.42 
.40 
.42 
.42 
.40 
.42 
.42 
.42 
.42 
.42 

.42 
.42 
.42 
.42 
.42 
.42 
.43 
.42 
.42 
.43 

.42 
.42 
.43 
.42 
.43 
.42 
.43 
.43 
.42 
.43 

.43 
.43 
.42 

.43 
.43 
.43 
.43 
.43 
.43 
.43 

.43 
.45 
.43 
.43 
.43 
.45 
.43 
.45 
.43 
.43 

.45 
.45 
.43 
.45 
.45 
.43 
.45 
.45 
.45 

D.r. 
180 



Tang. 



.289326 
.289999 
.290671 
.291342 
.292013 
.292682 
.293350 
.294017 
.294684 
.295349 

1.296013 
.296677 
.297339 
.298001 
.298662 
.299322 
.299980 



.301295 
.301951 

9.302607 
.303261 
.303914 
.304567 
.305218 



.306519 
.307168 
.307816 



9.309109 
.309754 
.310399 
.311042 
.311685 
.312327 
.312968 
.313608 
.314247 
.314885 

9.315523 
.316159 
.316795 
.317430 
.318064 
.318697 
.319330 
.319961 
.320592 
.321222 

9.321851 
.322479 
.323106 
.323733 

.324358 
.324983 
.325607 
.326231 
.326853 
9.327475 

Cotang. 



D. 1". 



11.23 
11.22 
11.20 
11.18 
11.18 
11.15 
11.13 
11.12 
11.12 
11.08 
11.07 

11.07 
11.03 
11.03 
11.02 
11.00 
10.97 
10.97 
10.95 
10.93 
10.93 

10.90 
10.88 
10.88 
10.85 
10.85 
10.83 
10.82 
10.80 
10.78 
10.77 

10.75 
10.75 
10.72 
10.72 
10.70 
10.68 
10.67 
10.65 
10.63 
10.63 

10.60 
10.60 
10.58 
10.57 
10.55 
10.55 
10.52 
10.52 
10.50 
10.48 

10.47 
10.45 
10.45 
10.42 
10.42 
10.40 
10.40 
10.37 
10.37 

D. 1'. 



Cotang. 



10.711348 
.710674 
.710001 
.709329 
.708658 
.707987 
.707318 
.706650 
.705983 
.705316 
.704651 

10.703987 
.703323 
.702861 
.701999 
.701338 
.700678 
.700020 
.699362 
.698705 
.698049 

10.697393 

.696739 
.696086 
.695433 
.694782 
.694131 
.693481 
.692832 
.692184 
.691537 

10.690891 
.690246 
.689601 
.688958 
.688315 
.687673 
.687032 
.686392 
.685753 
.685115 

10.684477 

.683841 
.683205 
.682570 
.681936 
.681303 
.680670 
.680039 
.679408 
.678778 

10.678149 
.677521 
.676894 
.676267 
.675642 
.675017 
.674393 
.673769 
.673147 

10.672525 

Tang. 



78* 



12° 



COSINES, TANGENTS, AND COTANGENTS. 167 . 



10 

11 

12 

13 
14 
15 
16 
17 
13 
19 
20 

n 

22 
23 
24 
25 
26 
27 
28 
29 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

51 
52 
53 
54 
55 
56 
57 
58 
59 



Sine. D. 1*. | Cosine. D. 1". Tang. D. 1". Cotang, 



9.317879 
.318473 
.319066 
.319658 
.320249 
.320840 
.321430 
.322019 
.322607 
.323194 
.323780 

9.324366 
.324950 
.325534 

.326117 
.326700 
.327281 
.327862 
.328442 
.329021 
.329599 

9.330176 
. -330753 
.331329 
.331903 
.332478 
.333051 
.333624 
.334195 
.334767 
.335337 

9.335906 
.336475 
.337043 
.337610 
.338176 
.338742 
.339307 
.339871 
.340434 
.340996 

9.341558 
.342119 
.342679 
.343239 
.343797 
.344355 
.344912 
.345469 
.346024 
.346579 

9.347134 

.347687 
.348240 
.348792 
.349343 
.349893 
.350443 
.350992 
.351540 
9.352088 



Cosine. 



9.90 
9.88 
9.87 
9.85 
9.85 
9.83 
9.82 



9.77 
9.77 

9.73 
9.73 
9.72 
9.72 
9.68 
9.68 
9.67 
9.65 
9.63 
9.62 

9.62 
9.60 
9.57 
9.58 
9.55 
9.55 
9.52 
9.53 
9.50 
9.48 

9.48 
9.47 
9.45 
9.43 
9.43 
9.42 
9.40 
9.38 
9.37 
9.37 

9.35 
9.33 
9.33 
9.30 
9.30 
9.28 
9.28 
9.25 
9.25 
9.25 

9.22 
9.22 
9.20 
9.18 
9.17 
9.17 
9.15 
9.13 
9.13 



9.990404 
.990378 
.990:351 
.990324 
.990297 
.990270 
.990243 
.990215 
.990188 
.990161 
.990134 

9.990107 
.990079 
.990052 
.990025 
.989997 
.989970 
.989942 
.989915 
.989887 



9.989832 
.989804 
.989777 
.989749 
.989721 
.989693 
.989665 
.989037 
.989610 



9.989553 
.989525 
.989497 
.989469 
.989441 
.989413 
.989385 



.989328 



9.989271 
.989243 
.989214 
.989186 
.989157 
.989128 
.989100 
.989071 
.989042 
.989014 

9.988985 



.988.927 



.988869 
.988840 



.988782 

.988753 

9.988724 



d. r 



Sine, 



.43 
.45 
.45 
.45 
.45 
.45 
.47 
.45 
.45 
.45 
.45 

.47 
.45 
.45 
.47 
.45 
.47 
.45 
.47 
.45 
.47 

.47 
.45 
.47 
.47 
.47 
.47 
.47 
.45 
.47 
.48 

.47 
.47 
.47 
.47 
.47 
.47 
.48 
.47 
.47 
.48 

.47 



.48 
.47 
.48 
.48 
.47 
.48 

.48 



.48 
.48 
.48 
.48 
.48 
.48 



d. r. 



9.327475 
.328095 
.328715 
.329334 
.329953 
.330570 
.331187 
.331803 
.332418 
.333033 
.333646 

9.334259 
.3:34871 
.3:35482 
.336093 
.336702 
.337311 
.337919 
.338527 
.339133 
.339739 

9.340344 

.340948 
.341552 
.342155 
.342757 
.343358 
.343958 
.344558 
.345157 
.345755 

9.346353 
.340949 
.347545 
.348141 
.348735 
.349329 
.349922 
.350514 
.351100 
.351697 

9.352287 
.352876 
.353-465 
.354053 
..354640 
.355227 
.355813 
.356398 
.356982 
.357566 

9.358149 

.358731 
.359313 
.359893 
.360474 
.361053 
.361632 
.362210 
.362787 
9.363364 



10.33 
10.33 
10.32 
10.32 
10.28 
10.28 
10.27 
10.25 
10.25 
10.22 
10.22 

10.20 
10.18 
10.18 
10.15 
10.15 
10.13 
10.13 
10.10 
10.10 
10.08 

10.07 

10.07 

10.05 

10.03 

10.02 

10.00 

10.00 

9.98 

9.97 

9.97 

9.93 
9.93 
9.93 
9.90 
9.90 
9.88 
9.87 
9.87 
9.85 
9.83 

9.82 
9.82 
9.80 
9.78 
9.78 
9.77 
9.75 
9.73 
9.73 
9.72 

9.70 
9.70 
9.67 
9.68 
9.65 
9.65 
9.63 
9.62 
9.62 



10.672525 60 
.671905 59 

.671285 58 

.670666 

.670047 



.668197 
.667582 
.666967 
.666354 

10.665741 
.665129 
.664518 
.663907 
.663298 
.662689 
.66^081 
.661473 
.660867 
.660261 



55 
54 
53 
52 
51 
50 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

10.659656 89 

.659052 38 

.658448 37 

.657845 36 

.657243 35 

.656642 34 

.656042 33 

.655442 32 

.654843 31 

.654245 30 

10.653647 29 

.653051 28 

.652455 27 

.651859 26 

.651265 25 

.650671 24 
.650078 | 23 

.649486 22 

.648894 21 

.648303 20 

19 

ia 

17 
16 
15 
14 
13 
12 
11 
10 

9 
8 

7 



10.647713 
.647124 
.646535 
.645947 
.645360 
.644773 
.644187 
.643602 
.643018 
.642434 

10.641851 
.641269 
.640687 
.640107 
.639526 
.6:38947 
.638368 
.637790 
.637213 

10.636636 



Cotang. | D. 1*. ! Tang. 



102° 



181 



7T° 



13° 



LOGARITHMIC SINES, 



166* 



, i 



10 

11 

12 
13 
14 
15 
16 
17 
18 
19 
20 

21 

22 
23 
24 
25 
26 
27 
28 
29 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

51 
52 
53 
54 
55 
56 
57 
58 
59 



X03° 



Sine. 



9.352088 
.352635 
.353181 
.353726 
.354271 
.354815 
.355358 
.355901 
.356443 
.356981 
.357524 

9.358064 
.358603 
.359141 
.359678 
.360215 
.360752 
.361287 
.361822 
.362356 



9.36^422 
.363954 
.364485 
.365016 
.365546 
.366075 
.366604 
.367131 
.367659 
.368185 

9.368711 



.369761 
.370285 
.370808 
.371330 
.371852 
.372373 
.372894 
.373414 

9.373933 

.374452 
.374970 
.375487 
.376003 
.376519 
.377035 
.377549 
.378063 
.378577 

9-379089 
.379601 
.380113 
.380624 
.381134 
.381643 
.382152 
.382661 



b»o 



Cosine. 



D. 1". 



9.12 
9.10 
9.08 
9.08 
9.07 
9.05 
9.05 
9.03 
9.02 
9.00 
9.00 

8.98 
8.97 
8.95 
8.95 
8.95 
8.92 
8.92 
8.90 



8.87 
8.85 
8.85 
8.83 
8.82 
8.82 
8.78 
8.80 
8.77 
8.77 

8.75 
8.75 
8.72 
8.72 
8.70 
8.70 



8.67 
8.65 

8.65 
8.63 
8.62 
8.60 
8.60 
8.60 
8.57 
8.57 
8.57 
8.53 

8.53 
8.53 
8.52 
8.50 
8.48 
8.48 
8.48 
8.45 
8.45 

d. r. 



Cosine. 



9.988724 



.988666 
.988636 



.988578 
.988548 
.988519 



.988430 

9.988401 
.988371 

.988342 
.988312 
.988282 
.988252 



.988133 

9.988103 
.988073 
.988043 
.988013 
.987983 
.987953 
.987922 
.987892 
.987862 
.987832 

9.987801 
.987771 
.987740 
.987710 
.987679 
.987649 
.987618 
.987588 
.987557 
.987526 

9.987496 

.987465 
.987434 
.987403 
.987372 
.987341 
.987310 
.987279 
.987248 
.987217 

9.987186 
.987155 
.987124 
.987092 
.987061 
.987030 



.986967 

.986936 

9.986904 

Sine. 



D. 1". 



.48 
.48 
.50 



.50 
.48 
.50 
.48 
.50 
.48 

.50 

.48 
.50 
.50 
.50 
.48 
.50 
.50 
.50 
.50 

.50 
.50 
.50 
.50 
.50 
52 
!50 
.50 
.50 
.52 

.50 
.52 
.50 
.52 
.50 
.52 
.50 
.52 
.52 
.50 

.52 
.52 
.52 
.52 
.52 
.52 
.52 
.52 
.52 
.52 

.52 
.52 
.53 
.52 

.52 
.53 

.52 
.52 
.53 



D. 1". 



182 



Tang. 



9.363364 
.363940 
.364515 
.365090 
.365664 
.366237 
.366810 
.367382 
.367953 
.368524 
.369094 



.370232 
.370799 
.371367 
.371933 
.372499 
.373064 
.373629 
.374193 
.374756 

9.375319 

.375881 
.376442 
.377003 
.377563 
.378122 
.378681 
.379239 
.379797 
.380354 

9.880910 
.381466 
.382020 
.382575 
.383129 
.383682 
.384234 
.384786 
.385337 
.385888 

9.386438 



.387536 



.388631 
.389178 
.389724 
.390270 
.390815 
.391360 

9.391903 
.392447 
.392989 
.393531 
.394073 
.394614 
.395154 
.395694 
.396233 

9.396771 

Cotang. 



D. 1'. 



9.58 
9.58 
9.57 
9.55 
9.55 
9.53 
9.52 
9.52 
9.50 
9.48 

9.48 
9.45 
9.47 
9.43 
9.43 
9.42 
9.42 
9.40 
9.38 
9.38 

9.37 
9.35 
9.35 
9.33 
9.32 
9.32 
9.30 
9.30 
9.28 
9.27 

9.27 
9.23 
9.25 
9.23 
9.22 
9.20 
9.20 
9.18 
9.18 
9.17 

9.15 
9.15 
9.13 
9.12 
9.12 
9.10 
9.10 
9.08 
9.08 
9.05 

9.07 
9.03 
9.03 
9.03 
9.02 
9.00 
9.00 
8.98 
8.97 

D. 1\ 



Cotang. 



10.636636 
.636060 
.635485 
.634910 
.634336 
.633763 
.633190 
.632618 
.632047 
.631476 
.630906 

10.630337 
.629768 
.629201 
.628633 
.628067 
.627501 
.626936 
.626371 
.625807 
.625244 

10.624681 
.624119 
.623558 
.622997 
.622437 
.621878 
.621319 
.620761 
.620203 
.619646 

10.619090 
.618534 
.617980 
.617425 
.616871 
.616318 
.615766 
.615214 
.614663 
.614112 

10.613562 
.613013 
.612464 
.611916 
.611369 
.610822 
.610276 
.609730 
.609185 
.608640 

10.608097 
.607553 
.607011 
.606469 
.605927 
.605386 
.604846 
.604306 
.603767 

10.603229 

Tang. 



76' 



COSINES, TANGENTS, AND COTANGENTS. 1Q5<3 



' Sine. D. 1\ Cosine. D. 1*. Tang. D. 1". Cotang, 



.384182 



.385192 



.3867'04 
.387207 
.387709 
.388210 
.388711 

9.389211 
.389711 
.390210 
.390708 
.391206 
.391703 
.392199 
.392695 
.393191 
.393685 

9.394179 
.394073 
.395106 
.395658 
.396150 
.396041 
.397132 
.397621 
.398111 
.398600 

9.399088 
.399575 
.400062 
.400549 
.401035 
.401520 
.402005 
.402489 
.402972 
.40&455 

9.403938 
.404420 
.404901 
.405382 
.405862 
.406341 
.406820 
.407299 
.407777 
.408254 

9.408731 
.409207 
.409682 
.410157 
.410632 
.411106 
.411579 
.412052 
.412524 

9.412996 



8.45 
8.42 
8.42 
8.42 
8.40 
8.38 
8.38 
8.37 
8.35 
8.35 
8.33 

8.33 
8.32 
8.30 
8.30 
8.23 
8.27 
8.27 
8.27 
8.23 
8.23 

8.23 

8.22 
8.20 
8.20 
8.18 
8.18 
8.15 
8.17 
8.15 
8.13 

8.12 
8.12 
8.12 
8.10 
8.08 
8.08 
8.07 
8.05 
8.05 
8.05 

8.03 
8.02 
8.02 
8.00 
7.98 
7.98 
7.98 
7.97 
7.95 
7.95 

7.93 
7.93 
7.92 
7.92 
7.90 
7.88 
7.88 
7.87 
7.87 



9.986904 
.986873 
.986841 
.986809 
.986778 
.986746 
.986714 



.986619 



9.986555 
.986523 

.986491 
.986459 
.986427 
.986395 



.936299 
.986266 

9.936234 
.986202 
.986169 
.986137 
.986104 
.986072 
.986039 
.986007 
.985974 
.985942 

9.985909 
.985876 
.985843 
.985811 
.985778 
.985745 
.985712 
.985679 
.985646 
.985613 

9.985580 
.985547 
.985514 
.985480 
.985447 
.985414 



.985347 
.985314 



9.985247 
.985213 
.985180 
.985146 
.985113 
.985079 
.985045 
.985011 
.984978 

9.984944 



.52 
.53 
.53 
.52 
.53 
.53 
.52 
.53 
.53 
.53 
.53 

.53 
.53 
.53 
.53 
.53 
.53 
.53 
.53 
.55 
.53 

.53 
.55 
.53 
.55 
.53 
.55 
.53 
.55 
.53 
.55 

.55 
.55 
.53 
.55 
.55 
.5") 
.55 
.55 



.55 

.55 
.57 
.55 
.55 
.55 
.57 
.55 
.57 
.55 

.57 
.55 
.57 
.55 
.57 
.57 
.57 
.55 
.57 



). 396771 
.397309 
.397846 
.398383 
.398919 
.399455 
.399990 
.400524 
.401058 
.401591 
.402124 

). 402656 
.403187 
.403718 
.404249 
.404778 
.405308 
.405336 
.406364 
.406392 
.407419 

1.407945 
.403471 



.409521 
.410045 
.410569 
.411092 
.411615 
.412137 
.412658 

9.413179 
.413699 
.414219 
.414738 
.415257 
.415775 
.416293 
.416810 
.417328 
.417842 

9.418358 
.418873 
.419387 
.419901 
.420415 
.420927 
.421440 
.421952 
.422463 
.422974 

9.42S484 
.423993 
.424503 
.425011 
.425519 



Cosine. I D. 1*. 



Sine. 



104° 



D. 1". I 
183" 



.426534 
.427041 
.427547 
.428052 



8.97 
8.95 
8.95 
8.93 
8.93 
8.92 
8.90 
8.90 
8.88 
8.88 
8.87 

8.85 
8.85 
8.85 
8.82 
8.83 
8.80 



8.75 
8.73 
8.73 
8.72 
8.72 
8.70 
8.68 
8.68 

8.67 
8.67 
8.65 
8.65 
8.63 
8.63 
8.62 
8.60 
8.00 
8.60 

8.58 
8.57 
8.57 
8.57 
8.55 
8.55 
8.53 
8.52 
8.52 
8.50 

8.48 
8.50 
8.47 
8.47 
8.47 
8.45 
8.45 
8.43 
8.42 



10.603229 
.602691 
.602154 
.601617 
.601081 
.600545 
.600010 
.599476 
.598942 
.598409 
.597876 

10.597344 
.596813 
.596282 
.595751 
.595222 
.594692 
.594164 
.593636 
.583108 
.592581 

10.592055 
.591529 
.591004 
.590479 
.589955 
.589431 



.588385 
.587863 
.587342 

10.586821 
.586301 
.585781 
.585262 
.584743 
.584225 
.583707 
.583190 
.582674 
.582158 

10.581642 
.581127 
.580613 
.580099 
.579585 
.579073 
.578560 
.578048 
.577537 
.577026 

10.576516 
.576007 
.575497 
.574989 
.574481 
.573973 
.573466 
.572959 
.572453 

10.571948 



59 
58 

57 
56 
55 
54 
53 
52 
51 
50 

49 
43 
47 
46 
45 
44 
43 
42 
41 
40 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 

19 
18 
17 
16 
15 
14 
13 
12 
11 
10 



Cotang. i D. 1". I Tang. 



75° 



15° 



LOGABITHMIG SINES, 



164° 



9 

10 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

21 

22 
23 
24 
25 

26 

27 
28 
23 
30 

31 
32 
S3 
34 
35 
36 
37 
38 
39 
40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



Sine. 



9.412996 
.413467 
.413938 
.414408 
.414878 
.415347 
.415815 
.416283 
.416751 
.417217 
.417684 

9.418150 
.418615 
.419079 
.419544 
.420007 
.420470 
.420933 
.421395 
.431857 
.422318 

9.422778 
.423238 
.423697 
.424156 
.424615 
.425073 
.425530 
425987 
.426443 
.426899 

9.427354 

.427809 
.423263 
.428717 
.429170 
.429623 
.430075 
.430527 
.430978 
.431429 

9.431879 
.432329 
.432778 
.433226 
.433675 
.434122 
.434569 
.435016 
.435462 
.435908 

9.436353 

.436798 
.437242 
.437686 
.438129 
.438572 
.439014 
.439456 
.439897 
9.440338 

Cosine. 



D. 1". 



7.85 
7.85 
7.83 
7.83 
7.82 
7.80 
7.80 
7.80 
7.77 
7.78 
7.77 

7.75 
7.73 
7.75 

7.72 
7.72 
7.72 
7.70 
7.70 
7.68 
7.67 

7.67 
7.65 
7.65 
7.65 
7.63 
7.62 
7.62 
7.60 
7.60 
7.58 

7.58 
7.57 
7.57 
7.55 
7.55 
7.53 
7.53 
7.52 
7.52 
7.50 

7.50 
7.48 
7.47 
7.48 
7.45 
7.45 
7.45 
7.43 
7.43 
7.42 

7.42 
7.40 
7.40 
7.38 
7.38 
7.37 
7.37 
7.35 
7.35 

D. 1\ 



Cosine. 



9.984944 
.984910 

.984876 
.984842 



.984774 
.984740 
.984706 
.984672 
.984638 
.984603 

9.984569 
.984535 
.984500 
.984466 
.984432 
.984397 
.984363 
.984328 
.984294 
.984259 

9.984224 
.984190 
.984155 
.984120 
.984085 
.984050 
.984015 
.983981 
.C83946 
.983911 

9.983875 
.983840 
.983805 
.983770 
.983735 
.983700 
.983664 
.983629 
.983394 
.983558 

9.983523 
.983487 
.983452 
.983416 
.983381 
.983345 
.983309 
.983273 
.983238 
.983202 

9.983166 
.983130 
.983094 



.983022 



.982950 
.982914 

.982878 



Sine. 



D. 1\ 



105° 



.57 
.57 
.57 
.57 
.57 
.57 
.57 
.57 
.57 
.58 
.57 

.57 

.58 
.57 
.57 
.58 
.57 
.58 
.57 
.58 
.58 

.57 
.58 
.58 
.58 
.58 
.58 
.57 
.58 
.58 
.60 

.58 
.58 
.58 
.58 
.58 
.60 
.58 
.58 
.60 
.58 

.60 
.58 
.60 
.58 
.60 
.60 
.60 
.58 
.60 
.60 

.60 
.60 
.60 
.60 
.60 
.60 
.60 
.60 
.60 

D. 1". 

184 



Tang, 



d. r 



9.428052 
.428558 
.429062 
.429566 
.430070 
.430573 
.431075 
.431577 
.432079 
.432580 
.433080 

9.433580 
.434080 
.434579 
.435078 
.435576 
.436073 
.436570 
.437067 

' .437563 
.438059 

9.438554 
.439048 
.439543 
.440036 
.440529 
.441022 
.441514 
.442006 
.442497 
.442988 

9.443479 
.443968 
.444458 
.444947 
.445435 
.445923 
.446411 
.446898 
.447384 
.447870 

9.448356 
.448841 
.449326 
.449810 
.450294 
.450777 
.451260 
.451743 
.452225 
.452706 

9.453187 
.453668 
.454148 
.454628 
.455107 
.455586 
.456064 
.456542 
.457019 

9.457496 

Cotang. 



8.43 
8.40 
8.40 
8.40 
8.38 
8.37 
8.37 
8.37 
8.35 
8.33 
8.33 

8.33 

8.32 
8.32 
8.30 
8.28 
8.28 
8.28 
8.27 
8.27 
8.25 

8.23 

8.25 
8.22 
8.22 
8.22 
8.20 
8.20 
8.18 
8.18 
8.18 

8.15 
8.17 
8.15 
8.13 
8.13 
8.13 
8.12 
8.10 
8.10 
8.10 

8.08 
8.08 
8.07 
8.07 
8.05 
8.05 
8.05 
8.03 
8.02 
8.02 

8.02 
8.00 
8.00 
7.98 
7.98 
7.97 
7.97 
7.95 
7.95 

D. 1". 



Cotang. 



10.571948 
.571442 
.570938 
.570434 
.569930 
.569427 
.568925 
.568423 
.567921 
.567420 
.566920 

10.566420 
.565920 
.565421 
.564922 
.564424 
.563927 
.563430 
.562933 
.562437 
.561941 

10.561446 

.560952 
.560457 
.559964 
.559471 
.558978 
.558486 
.557994 
.557503 
.557012 

10.556521 
.556032 
.555542 
.555053 
.554565 
.554077 
.553589 
.553102 
.552616 
.552130 

10.551644 
.551159 

.550674 
.550190 
.549706 
.549223 
.548740 
.548257 
.547775 
.547294 

10.546813 
.546332 

.545852 
.545372 
.544893 
.544414 
.543936 
.543458 
.542981 
10.542504 

Tang. 



74° 



16- 



COSINES, TANGENTS, AND COTANGENTS. 



163° 



39 
40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



Sine. 



9.440338 
.440778 
.441218 
.441658 
.442096 
.442535 
.442973 
.443410 
.443847 
.444284 
.444720 

9.445155 
.445590 
.446025 
.446459 
.446893 
.447326 
.447759 
.448191 
.448623 
.449054 

9.449485 
.449915 
.450345 
.450775 
.451204 
.451632 
.452060 
.452488 
.452915 
.453342 

9.453768 
.454194 
.454619 
.455044 
.455469 
.455893 
.456316 
.456739 
.457162 
.457584 

9.458006 
.458427 

.458848 



.460108 
.460527 
.460946 
.461364 
.461782 

9.462199 
.462616 
.463032 
.46:3448 
.463864 
.464279 
.464694 
.465108 
.465522 

9.465935 



D. 1". 



7.33 
7.33 
7.33 
7.30 
7.32 
7.30 
7.28 
7.28 
7.28 
7.27 
7.25 

7.25 
7.25 
7.23 
7.23 

7.22 
7.22 
7.20 
7.20 
7.18 
7.18 

7.17 
7.17 
7.17 



7.15 
7.13 
7.13 
7.13 
7.12 
7.12 
7.10 

7.10 



7. US 

7.07 
7.05 
7.05 
7.05 
7.03 
7.03 

7.02 
7.02 
7.00 
7.00 
7.00 



6.97 
6.97 
6.95 

6.95 
6.93 
6.93 
6.93 
6.92 
6.92 
6.90 
6.90 



Cosine. I D. 1". 



Cosine. 



'.982842 
.982805 



.982660 
.982624 
.982587 
.982551 
.982514 
.982477 

1.982441 
.982404 



.982331 



.982257 
.982220 
.982183 
.982146 
.982109 

9.982072 
.982035 



.981961 
.981924 
.981886 
.981849 
.981812 
.981774 
.981737 

9.981700 
.981662 
.981625 
.981587 
.981549 
.981512 
.981474 
.981436 



.981361 

9.981323 

.981285 
.981247 



.981171 
.981133 
.981095 
.981057 
.981019 
.980981 

9.980942 
.980904 



.980750 
.980712 
.980673 
.980635 



Sine. 



D. 1*. 



.62 
.60 
.60 
.62 
.60 
.60 
.62 
.60 
.62 
.62 
.60 

.62 
.62 
.60 
.62 
.62 
.62 
.62 
.62 
.62 
.62 

.62 
.62 
.62 
.62 
.63 
.62 
.62 
.63 
.62 
.62 

.63 

.62 
.63 
.63 
.62 
.63 
.63 
.62 
.63 
.63 

.63 
.63 
.63 
.63 
.63 
.63 
.63 
.63 
.63 
.65 

.63 
.63 
.65 
.63 
.65 
.63 
.65 
.63 
.65 



100° 



D. r. 

185 



Tang. 



L457496 
.457973 
.458449 
.458925 
.459400 
.459875 
.460349 
.460823 
.461297 
.461770 
.462242 

'.462715 
.463186 
.463658 
.464128 
.464599 
.465069 
.465539 
.466008 
.466477 
.466945 

1.467413 

.467880 
.468347 
.468814 



.469746 
.470211 

.470676 
.471141 
.471605 

1.472069 
.472532 
.472995 
.473457 
.473919 
.474381 
.474842 
.475303 
.475763 
.476223 

'.476683 
.477142 
.477601 
.478059 
.478517 
.478975 
.479432 
.479889 
.480345 



9.481257 
.481712 
.482167 
.482621 
.483075 
.483529 
.483982 
.484435 
.484887 

9.485339 



Cotang. 



D. 1". 



7.93 
7.93 
7.92 
7.92 
7.90 
7.90 
7.90 
7.88 
7.87 
7.88 

7.85 
7.87 
7.83 
7.85 
7.83 
7.83 
7.82 
7.82 



7.78 
7.78 
7.77- 
7.77 
7.75 
7.75 
7.75 
7.73 
7.73 

7.72 
7.72 
7.70 
7.70 
7.70 
7.68 
7.68 
7.67 
7.67 
7.67 

7.65 

7.65 
7.63 
7.63 
7.63 
7.62 
7.62 
7.60 
7.60 
7.60 

7.58 
7.58 
7.57 
7.57 
7.57 
7.55 
7.55 
7.53 
7.53 



D. 1". 



Cotang. 



10.542504 
.542027 
.541551 
.541075 
.540600 
.540125 
.539651 
.539177 
.538703 
.538230 
.537758 

10.537285 
.536814 
.536342 
.535872 
.535401 
.534931 
.534461 
.533992 
.533523 
.533055 

10.532587 
.532120 
.531653 
.531186 
.530720 
.530254 
.529789 
.529324 
.528859 
.528395 

10.527931 

.527468 
.527005 
.526543 
.526081 
.525619 
.525158 
.524697 
.524237 
.523777 

10.523317 

.522858 
.522399 
.521941 
.521483 
.521025 
.520568 
.520111 
.519655 
.519199 

10.518743 

.518288 
.517833 
.517379 
.516925 
.516471 
.516018 
.515565 
.515113 
10.514661 



Tang. 



60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

39 
38 
37 
36 
35 
34 
33 
32 
31 
SO 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 

19 
18 
17 
16 
15 
14 
13 
12 
11 
10 

9 
8 

6 
5 
4 
3 
2 
1 




73° 



17* 



LOGABITHMIC SINES, 



162* 



9 

10 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

21 
22 
23 
24 

25 
26 

27 
23 
29 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



Sine. 



9.465935 
.466348 
.466761' 
.467173 
.467585 
.467996 
.468407 
.468817 
.469227 
.469637 
.470046 

9.470455 
.470863 
.471271 
.471679 
.472086 
.472492 
.472398 
.473304 
.473710 
.474115 

9.474519 
.474923 
.475327 
.475730 
.476133 
.476536 
.476938 
.477340 
.477741 
.478142 

9.478542 
.478942 
.479342 
.479741 
.480140 
.480539 
.480937 
.481334 
.481731 
.482128 

9.482525 

.482921 
.483316 
.483712 
.484107 
.484501 
.484895 
.485289 
.485682 
.486075 

9.486467 

.486860 
.487251 
.487643 
.488034 
.488424 
.488814 
.489204 
.489593 
9.489982 

Cosine. 



d. r, 



6.87 
6.87 
6.85 
6.85 
6.83 
6.83 
6.83 
6.82 
6.82 

6.80 
6.80 
6.80 
6.78 
6.77 
6.77 
6.77 
6.77 
6.75 
6.73 

6.73 
6.73 
6.72 
6.72 
6.72 
6.70 
6.70 
6.68 
6.68 
6.67 

6.67 
6.67 
6.65 
6.65 
6.65 
6.63 
6.62 
6.62 
6.62 
6.62 

6.60 
6.58 
6.60 
6.58 
6.57 
6.57 
6.57 
6.55 
6.55 
6.53 

6.55 
6.52 
6.53 
6.52 
6.50 
6.50 
6.50 
6.48 
6.48 

D 1\ 



Cosine. 



.980325 
.980286 
.980247 



Sine. 



D. 1\ 



.65 
.65 
.63 
.65 
.65 
.65 
.65 
.65 
.65 
.65 

.65 
.65 
.65 
.67 
.65 
.65 
.65 
.67 
.65 
.67 

.65 
.67 
.65 
.67 
.65 
.67 
.67 
.67 
.65 
.67 

.67 
.67 
.67 
.67 
.67 
.67 
.67 
.68 
.67 
.67 

.67 
.68 
.67 
.68 
.67 
.67 
.68 
.68 



.68 



.68 



.68 
.68 



107 9 



D. 1" 



186 



Tang. 



D. 1\ 



9.485339 
.485791 
.486242 
.486693 
.487143 
.487593 
.488043 
.48S492 
.488941 



9.490286 
.490733 
.491180 
.491627 
.492073 
.492519 
.492965 
.493410 
.493854 
.494299 

9.494743 
.495186 
.495630 
.496073 
.496515 
.496957 
.497399 
.497841 
.498282 
.498722 

9.499163 
.499603 
.500042 
.500481 
.500920 
.501359 
.501797 
.502235 
.502672 
.503109 

9.503546 
.503982 
.504418 
.504854 
.505289 
.505724 
.506159 
.506593 
.507027 
.507460 

9.507893 
.508326 
.508759 
.509191 
.509622 
.510054 
.510485 
.510916 
.511346 

9.511776 

Cotang. 



7.53 
7.52 

7.52 
7.50 
7.50 
7.50 

7.48 
7.48 
7.48 
7.47 
7.47 

7.45 
7.45 
7.45 
7.43 
7.43 
7.43 
7.42 
7.40 
7.42 
7.40 

7.38 
7.40 
7.38 
7.37 
7.37 
7.37 
7.37 
7.35 
7.33 
7.35 

7.33 
7.32 
7.32 
7.32 
7.32 
7.30 
7.30 
7.28 
7.28 
7.28 

7.27 
7.27 
7.27 
7.25 
7.25 
7.25 
7.23 
7.23 
7.23 
7.22 

7.22 
7.22 
7.20 
7.18 
7.20 
7.18 
7.18 
7.17 
7.17 

D. 1\ 



Cotang. 



10.514661 
.514209 
.513758 
.513307 
.512857 
.512407 
.511957 
.511508 
.511059 
.510610 
.510162 

10.509714 
.509267 
.508820 
.508373 
.507927 
.507481 
.507035 
.506590 
.506146 
.505701 

10.505257 
.504814 
.504370 
.503927 
.503485 
.503043 
.502601 
.502159 
.501718 
.501278 

10.500837 
.500397 
.499958 
.499519 
.499080 
.498641 
.498203 
.497765 
.497328 
.496891 

10.496454 
.496018 
.495582 
.495146 
.494711 
.494276 
.493841 
.493407 
.492973 
.492540 

10.492107 
.491674 
.491241 
.490809 
.490378 
.489946 
.489515 
.489084 
.488654 

10.488224 

Tang. 



72* 



18o COSINES, TANGENTS, AND COTANGENTS. 161o 



' Sine. 1X1". Cosine, D. 1*. Tang. D. 1". Cotang. 



9.489982 
.490371 
.490759 
.491147 
.491535 
.491922 



.492695 
.493081 
.493466 
.493851 

9.494236 
.494621 
.495005 
.495388 
.495772 
.496154 
.496537 
.496919 
.497301 
.497682 

9.498064 
.498444 



.499204 
.499584 
.499963 
.500342 
.500721 
.501099 
.501476 

9.501854 
.502231 
.502607 
.502984 
.503360 
.503735 
.504110 
.504485 
.504860 
.505234 

9.505608 
.505981 
.506354 
.506727 
.507099 
.507471 
.507843 
.508214 
.508585 
.508956 

9.509326 
.509696 
.510065 
.510434 
.510803 
.511172 
.511540 
.511907 
.512275 

9.512642 



6.48 
6.47 
6.47 
6.47 
6.45 
6.43 
6.45 
6.43 
6.42 
6.42 
6.42 

6.42 
6.40 
6.38 
6.40 
6.37 
6.38 
6.37 
6.37 
6.35 
6.35 

6.33 
6.35 
6.32 
6.33 
6.32 
6.32 
6.32 
6.30 
6.28 
6.30 

6.28 
6.27 
6.28 
6.27 
6.25 
6.25 
6.25 
6.25 
6.23 
6.23 

6.22 
6.22 
6.22 
6.20 
6.20 
6.20 
6.18 
6.18 
6.18 
6.17 

6.17 
6.15 
6.15 
6.15 
6.15 
6.13 
6.12 
6.13 
6.12 



Cosine. 



D. V 



9.978206 
.978165 
.978124 
.978083 
.978042 
.978001 
.977959 
.977918 
.977877 
.977835 
.977794 



9.977752 
.977711 
.977669 
.977628 
.977586 
.977544 
.977503 
.977461 
.977419 
.977377 

9.977335 

.977293 
.977251 
.977209 
.977107 
.977125 
.977083 
.977041 
.976999 
.976957 

9.976914 
.976872 
.976830 
.976787 
.976745 
.976702 
.976660 
.976017 
.976574 
.976532 

9.976489 
.976440 
.976404 
.976361 
.976318 
.976275 
.976232 
.976189 
.976146 
.976103 

9.976060 
.976017 
.975974 
.975930 
.975887 
.975844 
.975800 
.975757 
.975714 

9.975670 



.68 
.68 



.68 
.68 
.70 



.70 
.68 
.70 

.68 
.70 
.68 
.70 
.70 
.68 
.70 
.70 
.70 
.70 

.70 
.70 
.70 
.70 
.70 
.70 
.70 
.70 
.70 
.72 

.70 

.70 
72 
'.70 
.72 
.70 
.72 
.72 
.70 



.70 

.72 
.72 
.72 
.72 
.72 
.72 
.72 
.72 

.72 
.72 
.73 
72 
!72 
.7^ 
.72 
.72 



Sine. 



108° 



D. 1". 



187 



9.511776 
.512206 
.512635 
.513064 
.513493 
.513921 
.514349 
.514777 
.515204 
.515631 
.516057 

9.516484 
.516910 
.517335 
.517761 
.518186 
.518610 
.519034 
.519458 
.519882 
.520305 

9.520728 
.521151 
.521573 
.521995 
.522417 
.522338 
.523259 
.523680 
.524100 
.524520 

9.524940 
.525359 

.525778 
.526197 
.526615 
.527033 
.527451 
.527868 
.528285 
.528702 

9.529119 
.529535 
.529951 



.530781 
.531196 
.531611 
.532025 
.532439 
.532853 

9.533266 
.533679 
.534092 
.534504 
.534916 
.535328 
.535739 
.536150 
.536561 

9,536972 



7.17 
7.15 
7.15 
7.15 
7.13 
7.13 
7.13 
7.12 
7.12 
7.10 
7.12 

7.10 
7.08 
7.10 
7.08 
7.07 
7.07 
7.07 
7.07 
7.05 
7.05 

7.05 
7.03 
7.03 
7.03 
7.02 
7.02 
7.02 
7.00 
7.00 
7.00 

6.98 



6.97 
6.97 
6.97 
6.95 
6.95 
6.95 
6.95 

6.93 
6.93 
6.92 
6.92 
6.92 
6.92 
6.90 
6.90 
6.90 
6.88 

6.88 
6.88 
6.87 
6.87 
6.87 
6.85 
6.85 
6.85 
6.85 



Cotang. I D. 1". 



10.488224 
.487794 
.487365 
.486936 
.486507 
.486079 
.485651 
.485223 
.484796 
.484369 
.483943 

10.483516 
.483090 
.482665 
.482239 
.481814 
.481390 
.480966 
.480542 
.480118 
.479695 

10.479272 
.478849 
.478427 
.478005 
.477583 
.477162 
.476741 
.476320 
.475900 
.475480 

10.475060 
.474641 
.474222 
.473803 
.473385 
.472967 
.472549 
.472132 
.471715 
.471298 

10.470881 
.470465 
.470049 
.469634 
.469219 
.468804 
.468389 
.467975 
.467561 
.467147 

10.466734 
.466321 



.465496 
.465084 
.464672 
.464261 
.463850 
.463439 
10.463028 



Tang. 



59 
58 
57 
56 
55 
54 
53 
52 
51 
50 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 

29 
28 

27 
26 
25 
24 
23 
22 
21 
20 

19 
18 
17 
16 
15 
14 
13 
12 
11 
10 



71° 



19° 



LOGARITHMIC SINES, 



160« 



Sine. D. 1". Cosine. D. 1". Tang. D. 1 




1 
2 
3 
4 
5 
6 
7 
S 
9 
10 

11 

12 
13 
14 
15 
16 
17 
18 
19 
20 

21 
22 

23 
24 
25 
26 

27 
28 
29 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

51 
52 
53 
54 
55 
56 
57 
58 
59 
CO 



9.512642 
.513009 
.513375 
.513741 
.514107 
.514472 
.514837 
.515202 
.515566 
.515930 
.516294 

9.516657 
.517020 
.517382 
.517745 
.518107 
.518468 
.518829 
.519190 
.519551 
.519911 

9.520271 

.520631 
.520990 
.521349 
.521707 
.522066 
.522424 
.522781 
.523133 
.523495 

9.523852 

.524208 
.524564 
.524920 
.525275 
.525630 
.525984 
.526339 
.526693 
.527046 

9.527400 
.527753 
.528105 
.528458 
.528810 
.529161 
.529513 
.529864 
.530215 
.530565 

9.530915 
.531265 
.531614 
.531963 
.532312 
.532661 
.533009 
.533357 
.533704 

9.534052 



Cosine. D. 1 



6.12 
6.10 
6.10 
6.10 
6.08 
6.08 
6.08 
6.07 
6.07 
6.07 
6.05 

6.05 
6.03 
6.05 
6.03 
6.02 
6.02 
6.02 
6.02 
6.00 
6.00 

6.00 
5.98 
5.98 
5.97 
5.98 
5.97 
5.95 
5.95 
5.95 
5.95 

5.93 
5.93 
5.93 
5.92 
5.92 
5.90 
5.92 
5.90 
5.88 
5.90 

5.88 
5.87 
5.88 
5.87 
5.85 
5.87 
5.85 
5.85 
5.83 
5.83 

5.83 
5.82 
5.82 
5.82 
5.82 
5.80 
5.80 
5.78 
5.80 



9.975670 
.975627 
.975583 
.975539 
.975496 
.975452 
.975408 
.975365 
.975321 
.975277 
.975233 

9.975189 
.975145 
.975101 
.975057 
.975013 
.974969 
.974925 
.974880 
.974836 
.974792 

9.974748 
.974703 
.974659 
.974614 
.974570 
.974525 
.974481 
.974436 
.974301 
.974347 

9.974302 
.974257 
.974212 
.974167 
.974122 
.974077 
.974032 
.973987 
.973942 
.973897 

9.973852 
.973807 
.973761 
.973716 
.973671 
.973625 
.973.580 
.973535 
.973489 
.973444 

9.973398 
.973352 
.973307 
.973261 
.973215 
.973169 
.973124 
.973078 
.973032 

9.972986 



.72 
.73 
.13 
.72 
.73 
.73 
.72 
.73 
.73 
.73 
.73 

.73 
.73 
.73 
.73 



.73 
.73 
.73 

.75 
.73 
.75 
.73 
.75 
.73 

• 7 5 

.75 
.73 
.75 



.77 
.75 
.75 
.77 
.75 
.75 
.77 



.77 
.75 

.77 
.77 
.77 
.75 
.77 
.77 
.77 



Sine. 



D. 1", 



9.536972 
.537382 
.537792 
.538202 
.538611 
.539020 
.539429 
.539837 
.540245 
.540653 
.541061 

9.541468 
.541875 
.542281 
.542688 
.543094 
.543499 
.543905 
.544310 
.544715 
.545119 

9.545524 

.545928 
,546331 
. 5467S5 
.547138 
.547540 
.547943 
.548345 
.548747 
.549149 

9.549550 
.549951 
.550352 
.550752 
.551153 
.551552 
.551952 
.552351 
. 552750 
.553149 

9.553548 
.553946 
.554344 
.554741 
.555139 
.55.* 536 
.555933 
.556329 
. 556725 
.557121 

9.557517 
.557913 
.558308 
.558708 
.559097 
.559491 
.559885 
.560279 
.560673 

9.561066 



Cotang. 



6.82 
6.82 
6.82 
6.80 
6.80 
6.80 
6.80 
6.78 

6.78 
6.77 
6.78 
6.77 
6.75 
6.77 
6.75 
6.75 
6.73 
6.75 

6.73 
6.72 
6.73 
6.72 
6.70 
6.72 
6.70 
6.70 
6.70 
6.68 

6.68 
6.68 
6.67 
6.68 
6.65 
6.67 
6.65 
6.65 
6.65 
6.65 

6.63 
6.63 
6.62 
6.63 
6.62 
6.62 
6.60 
6.60 
6.60 
6.60 

6.60 
6.58 
6.58 
6.57 
6.57 
6.57 
6.57 
6.57 
6.55 



D. 1". 



Cotang. 



10.463028 
.462618 
.462208 
.461798 
.461389 
.460980 
.460571 
.460163 
.459755 
.459347 
.458939 

10.458532 
.458125 
.457719 
.457312 
.456906 
.456501 
.456095 
.455690 
.455285 
.454881 

10.454476 
.454072 
.453669 
.453265 
.452862 
.452460 
.452057 
.451655 
.451253 
.450851 

10.450450 
.450049 
.449648 
.449248 
.448847 
.448448 
.448048 
.447649 
.447250 
.446851 

10.446452 
.446054 
.445656 
.445259 
.444861 
.444464 
.444067 
.443671 
.443275 
.442879 

10.442483 

.4420S7 
.441692 
.441297 
.440903 
.440509 
.440115 
.439721 
.439327 
10.438931 



Tang. 



109* 



188 



70» 



Q6 COSINES, TANGENTS, AND COTANGENTS. lg9# 



1 


Sine. 


D. 1". 


Cosine. 


D. 1\ 


Tang. 


D. 1\ 


Cotang. 


/ 





9.534052 


5.78 
5.77 
5.78 
5.77 
5.75 
5.77 
5.75 
5.73 
5.75 
5.73 
5.73 


9.972986 


.77 

.77 


9.561066 


6.55 
6.53 
6.55 
6.53 
6.53 
6.52 
6.53 
6.52 
6.52 
6.50 
6.50 


10.438934 


60 


1 


.534399 


i .972940 


.561459 


.438541 


59 


2 


.534745 


! .972894 


.561851 


.438149 


58 


3 


.535092 


! .972848 


. i < 

.77 
.78 
.77 
.77 
.77 
.78 
.77 
.77 


.562244 


.437756 


57 


4 


.535438 


1 .972802 


.562636 


.437364 


56 


5 


.535783 


.972755 


.563028 


.436972 


55 


6 


.536129 


.972709 


.563419 


.436581 


54 


7 


.536474 


.972663 


.563811 


.436189 


53 


8 


.536818 


.972617 


.564202 


.435798 


52 


9 


.537163 


.972570 


.564593 


.435407 


51 


10 


. 537507' 


.972524 


.564983 


.435017 


50 


11 


9.537851 


5.72 
5.70 
5.70 
5.72 
5.70 
5.70 
5.68 
5.68 
5.68 
5.68 


9.972478 


.78 
.77 
.78 
.78 
.77 
.78 
.78 
.77 
.78 
.78 


9.565373 


6.50 
6.50 
6.48 
6.50 
6.47 
6.48 
6.48 
6.47 
6.45 
6.47 


10.434627 


49 


12 


.538194 


.972431 


.565763 


.434237 


48 


13 


.538538 


.972385 


.566153 


.433847 


47 


14 


.538880 


.972338 


.566542 


.433458 


46 


15 


.539223 


.972291 


.566932 


.433068 


45 


16 


.539565 


.972245 


.567320 


.4326S0 


44 


17 


.539907 


.972198 


.567709 


.432291 


43 


18 


.£40249 


.972151 


.568098 


.431902 


42 


19 


.540590 


.972105 


.568186 


.431514 


41 


20 


.540931 


.972058 


.568873 


.431127 


40 


21 


9.541272 


5.68 
5.67 
5.67 
5.65 
5.65 
5.65 
5.65 
5.63 
5.63 
5.63 


9.972011 


.78 
.78 
.78 
.78 
.78 
.78 
.78 
.78 
.78 
.80 


9.569261 


6.45 
6.45 
6.45 
6.45 
6.43 
6.43 
6.43 
6.42 
6.43 
6.42 


10.430739 


39 


22 


.541613 


.971964 


.569648 


.430352 


38 


23 


.541953 


.971917 


.570035 


.429965 


37 


24 


.542293 


.971870 


.570422 


.429578 


36 


25 


.542632 


.971823 


.570809 


.429191 


35 


26 


.542971 


.971776 


.571195 


.428805 


34 


27 


.543310 


.971729 


.571581 


.428419 


33 


28 


.543649 


.971682 


.571967 


.428033 


32 


29 


.543987 


.971635 


.572352 


.427648 


31 


30 


.544325 


.971588 


.572738 


.427262 


30 


31 


9.544663 


5.62 
5.63 
5.60 
5.62 
5.60 
5.60 
5.60 
5.58 
5.58 
5.58 


9.971540 


.78 
.78 
.60 
.78 
.80 
.78 
.80 
.78 
.80 
.78 


9.573123 


6.40 

6.42 
6.40 
6.40 
6.40 

6.38 
6.38 
6.38 
6.38 
6.38 


10.426877 


29 


32 


.545000 


.971493 


.573507 


.426493 


28 


33 


.545338 


.971446 


.573892 


.426108 


27 


34 


.545674 


.971398 


.574276 


.425724 


26 


35 


.546011 


.971351 


.574660 


.425340 


25 


36 


.546347 


.971303 


.575044 


.424956 


24 


37 


.546683 


.971256 


.575427 


.424573 


23 


38 


.547019 


.971208 


.575810 


.424190 


22 


39 


.54735-1 


.971161 


.576193 


.4238C7 


21 


40 


.547689 


.971113 


i .576576 


.423424 


20 


41 


9.548024 


5.58 
5.57 
5.57 
5.55 
5.55 
5.55 
5.55 
5.55 
5.53 
5.53 


9.971066 


.80 
.80 
.80 
• .80 
.78 
.80 
.80 
.80 
.80 
.82 


9.576959 


6.37 
6.37 


10.423041 


19 


42 


.548359 


.971018 


.577341 


.422659 


18 


43 


.548693 


.970970 


.577723 


.422277 


17 


44 


.549027 


.970922 


.578104 


6.'37 
6.35 
6.35 
6.35 
6.33 
6.33 
6.33 


.421896 


16 


45 


.549360 


.970874 


.578486 


.421514 


15 


46 


.549693 


.970827 


.578867 


.421133 


14 


47 


.550026 


.970779 


.579248 


.420752 


13 


48 


.550359 


.970731 


.579629 


.420371 


12 


49 


.550692 


.970683 


.580009 


.419991 


11 


50 


.551024 


.970635 


.580389 


.419611 


10 


51 


9.551356 


5.52 
5.52 
5.52 
5.52 
5.50 
5.52 
5.48 
5.50 
5.48 


9.970586 


.80 
.80 
.80 
.80 
.82 
.80 
.80 
.82 
.80 


9.580769 


6.33 
6.32 
6.32 
6.32 
6.32 
6.32 
6.30 
6.30 
6.28 


10.419231 


9 


52 


.551687 


.970538 


.581149 


.418851 


8 


53 


.552018 


.970490 


.5S1528 


.418472 


7 


54 


.552349 


.970442 


.581907 


.418093 


6 


55 


.552680 


.970394 


.582286 


.417714 


5 


56 


.553010 


.970345 


.582665 


.417335 


4 


57 


.553341 


.970297 


.583044 


.416956 


3 


58 


.553670 


.970249 


.583422 


.416578 


2 


59 


.554000 


.970200 


.583800 


.416200 


1 


60 


9.554329 


9.970152 


9.584177 


10.415823 





' 


Cosine. 1 


D. 1". i 


Sine. | 


d. r. l| 


Cotang. 


D.r. | 


Tang. 


' 1 



110° 



189 



QV 



21° 



LOGARITHMIC SINES, 



158° 



Sine. 



9.554329 
.554658 
.554987 
.555315 
.555643 
.555971 
.556299 
.556626 
.556953 
.557280 
.557606 

9.557932 

.558258 
.558583 
.558909 
.559234 
.559558 
.559883 
.560207 
.560531 
.560855 

9.561178 
.561501 
.561824 
.562146 
.562468 
.562790 
.563112 
.563433 
.563755 
.564075 

9.564396 
.564716 
.565036 
.565356 
.565676 
.565995 
.566314 
.566632 
.566951 
.567269 

9.567587 
.567904 
.568222 
.568539 
.568856 
.569172 
.569488 
.569804 
.570120 
.570435 

9.570751 
.571066 
.571380 
.571695 
.572009 
.572323 
.572636 
.572950 
.573263 

9.573575 



D. 1". 



Cosine. D. 1". 



5.48 
5.48 
5.47 
5.47 
5.47 
5.47 
5.45 
5.45 
5.45 
5.43 
5.43 

5.43 
5.42 
5.43 
5.42 
5.40 
5.42 
5.40 
5.40 
5.40 
5.38 

5.38 
5.38 
5.37 
5.37 
5.37 
5.37 
5.35 
5.37 
5.33 
5.35 

5.33 
5.33 
5.33 
5.33 
5.32 
5.32 
5.30 
5.32 
5.30 
5.30 

5.28 
5.30 
5.28 
5.28 
5.27 
5.27 
5.27 
5.27 
5.25 
5.27 

5.25 
5.23 
5.25 
5.23 
5.23 
5.22 
5.23 
5.22 
5.20 



Cosine. 



9.970152 
.970103 
.970055 
.970006 
.969957 
.969909 



.969811 
.969762 
.969714 
.969665 

9.969616 
.969567 
.969518 
.969469 
.969420 
.969370 
.969321 
.969272 
.969223 
.969173 

9.969124 
.969075 
.969025 
.968976 
.968926 
.968877 



.968777 
.968728 
.968678 

9.968628 
.968578 
.968528 
.968479 
.968429 
.968379 
.968329 
.968278 
.968228 
.968178 

9.968128 
.968078 
.968027 
.967977 
.967927 
.967876 
.967826 
.967775 
.967725 
.967674 

9.967624 
.967573 
.967522 
.967471 
.967421 
.967370 
.967319 
.967268 
.967217 

9.967166 



Sine. 



D. 1". 



82 



111° 



D. 1". 
190 



Tang. 



9.584177 
.584555 
.584932 
.585309 
.585686 
.586062 
.586439 
.586815 
.587190 
.587566 
.587941 

9.588316 
.588691 
.589066 
.589440 
.589814 
.590188 
.590562 
.590935 
.591308 
.591681 

9.592054 
.592426 
.592799 
.593171 
.593542 
.593914 
.594285 
.594656 
.595027 
.595398 

9.595768 
.596138 
.596508 
.596878 
.597247 
.597616 
.597985 
.598354 
.598722 
.599C91 

9.599459 

.599827 
.600194 
.600562 



.601296 
.601663 
.602029 
.602395 
.602761 

9.603127 
.603493 



.604223 
.604588 
.604953 
.6C5317 



.606046 
9.606410 



Cotang. 



D. 1". 



6.30 
6.28 
6.28 
6.28 
6.27 
6.28 
6.27 
6.25 
6.27 
6.25 
6.25 

6.25 
6.25 
6.23 
6.23 
6.23 
6.23 
6.22 
6.22 
6.22 
0.22 

G.20 
6.22 
6.20 
6.18 
6.20 
6.18 
6.18 
6.18 
6.18 
6.17 

6.17 
6.17 
6.17 
6.15 
6.15 
6.15 
6.15 
6.13 
6.15 
6.13 

6.13 
6.12 
6.13 
6.12 
6.12 
6.12 
6.10 
6.10 
6.10 
6.10 

6.10 



6.08 
6.08 
6.07 
6.08 
6.07 
6.07 



D. 1". 



Cotang. 



10.415823 
.415445 
.415068 
.414691 
.414314 
.413938 
.413561 
.413185 
.412810 
.412434 
.412059 

10.411684 
.411309 
.410934 
.410560 
.410186 
.409812 
.409438 
.409065 
.408692 
.408319 

10.407946 
.407574 
.407201 
.406829 
.406458 
.406086 
.405715 
.405344 
.404973 
.404602 

10.404232 
.403862 
.403492 
.403122 
.402753 
.402384 
.402015 
.401646 
.401278 
.400909 

10.400541 
.400173 
.399806 
.399438 
.399071 
.398704 
.398337 
.397971 
.397605 
.397239 

10.396873 
.396507 
.396142 
.395777 
.395412 
.395047 
.394683 
.394318 
.393954 

10.393690 



Tang. 





63 # 



COSINES, TANGENTS, AND COTANGENTS. lg7o 



Sine. D. 1". j Cosine. D. 1". Tang. D. 1". Cotang. 



9.57&57S 
.573888 
.574200 
.574512 
.574824 
.575136 
.575447 
.575758 
.576069 
.576379 
.576689 

9.576999 
.577309 
.577618 
.577927 
.578236 
.578545 
.578853 
.579162 
.579470 
.579777 

9.580085 
.580392 



.581005 
.581312 
.581618 
.581924 
.582229 
.582535 
.582840 

9.583145 
.583449 
.583754 
.584058 
.584361 
.584665 
.584968 
.585272 
.585574 
.585877 

9.586179 

.586482 
.586783 
.587085 
.587386 
.587688 
.587989 
.588289 
.588590 



9.589190 
.589489 
.589789 
.590088 
.590387 
.590686 
.590984 
.591282 
.591580 

9.591878 



! Cosine. 



5.22 
5.20 
5.20 
5.20 
5.20 
5.18 
5.18 
5.18 
5.17 
5.17 
5.17 

5.17 
5.15 
5.15 
5.15 
5.15 
5.13 
5.15 
5.13 
5.12 
5.13 

5.12 
5.12 
5.10 
5.12 
5.10 
5.10 
5.08 
5.10 
5.08 
5.08 

5.07 
5.08 
5.07 
5.05 
5.07 
5.05 
5.07 
5.03 
5.05 
5.03 

5.05 
5.02 
5.03 
5.02 
5.03 
5.02 
5.00 
5.02 
5.00 
5.00 

4.98 
5.00 
4.98 
4.98 
4.98 
4.97 
4.97 
4.97 
4.97 



9.967166 
.967115 
.967064 
.967013 
.966961 
.966910 
.966859 
.966808 
.966756 
.966705 
.966653 

9.966602 
.966550 
.966499 

.966447 



D. 1" 



.966344 
.966292 
.966240 
.966188 
.966136 

9.966085 
.966033 
.965981 
.965929 
.96.5876 
.965824 
.965772 
.965720 
.965668 
.965615 

9.965563 
.965511 
.965458 
.965406 
.965353 
.965301 
.965248 
.965195 
.965143 
.965090 

9.965037 
.964984 
.964931 
.964879 
.964826 
.964773 
.964720 



.964613 
.964560 

9.964507 
.964454 
.964400 
.964347 
.964294 
.964240 
.964187 
.964133 



9.964026 



Sine. 



87 



87 



90 



112' 



D. 1". 

191 



9.606410 
.606773 
.607137 
.607500 
.607863 
.608225 
.608588 
.608950 
.609312 
.609674 
.610036 

9.610397 
.610759 
.611120 
.611480 
.611841 
.612201 
.612561 
.612921 
.613281 
.613641 

9.614000 
.614359 
.614718 
.615077 
.615435 
.615793 
.616151 
.616509 
.616867 
.617224 

9.617582 
.617939 
.618295 
.918652 
.619008 
.619364 
.619720 
.620076 
.620432 
.620787 

9.621142 
.621497 
.621852 
.622207 
.622561 
.622915 
.623269 
.623623 
.623976 
.624330 

9.624683 
.625036 
.625388 
.625741 
.626093 
.626445 
.626797 
.627149 
.627501 

9.627852 



6.05 
6.07 
6.05 
6.05 
6.03 
6.05 
6.03 
6.03 
6.03 
6.03 
6.02 

6.03 
6.02 
6.00 
6.02 
6.00 
6.00 
6.00 
6.(0 
6.00 
5.98 

5.98 
5. 88 
5.98 
5.97 
5.97 
5.9r 
5.97 
5.97 
5.95 
5.97 

5.95 
5.93 
5.95 
5.93 
5.93 
5.93 
5.93 
5.93 
5.92 
5.92 

5.92 
5.92 
5.92 
5.90 
5.90 
5.90 
5.90 
5.88 
5.90 
5.88 

5.88 

5.87 
5.88 
5.87 
5.87 
5.87 
5.87 
5.87 
5.85 



10.393590 
.393227 



.392500 
.392137 
.391775 
.391412 
.391050 
.390688 
.390326 
.389964 

10.389603 
.389241 
.888880 
.388520 
.388159 
.387799 
.387439 
.387079 
.386719 
.386359 

10.386000 
.385641 



10.378858 
.378503 
.378148 
.377793 
.377439 
.377085 
.376731 
.376377 
.376024 
.375670 

10.375317 
.374964 
.374612 
.374259 
.373907 
.373555 
.373203 
.372851 
.372499 

10.372148 



.384923 
.384565 
.384207 
.383849 

.383491 32 

.383133 31 

.382776 30 

10.382418 29 

.382061 28 

.381705 27 

.381348 26 
.380992 | 25 
.380636 I 24 

.380280 23 

.379924 22 

.379568 21 

.379213 20 



Cotang. j D. 1" 



Tang. 



67° 



23° 



LOGARITHMIC SINES, 



156° 



o 
l 

2 
3 
4 
5 
6 
7 
8 
9 
10 

11 
12 
13 
14 
15 
16 
IT 
18 
19 
20 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

51 

52 
53 
54 
55 
56 
57 
58 
59 
60 



113» 



Sine. 



9.591878 
.592176 
.592473 
.592770 
.593067 
.593363 
.593659 
.593955 
.594251 
.594547 
.594842 

9.595137 
.595432 
.595727 
.596021 
.596315 
.596609 
.596903 
.597196 
.597490 
.597783 

9.598075 



.598660 
.598952 
.599244 
.599536 
.599827 
.600118 
.600409 
.600700 

9.600990 
.601280 
.601570 
.601860 
.602150 
.602439 
.602728 
.603017 
.603305 
.603594 

9.603882 
.604170 
.604457 
.604745 
.605032 
.605319 
.605606 
.605892 
.606179 
.606465 

9.606751 

.607036 
.607322 
.607607 
.607892 
.608177 
.608461 
.608745 
.609029 
9.609313 

Cosine. 



D. 1\ 



4.97 
4.95 
4.95 
4.95 
4.93 
4.93 
4.93 
4.93 
4.93 
4.92 
4.92 

4.92 
4.92 
4.90 
4.90 
4.90 
4.90 
4.83 
4.90 
4.88 
4.87 

4.88 
4.87 
4.87 
4.87 
4.87 
4.85 
4.85 
4.85 
4.85 
4.83 

4.83 
4.83 
4.83 
4.83 
4.82 
4.82 
4.82 
4.80 
4.82 
4.80 

4.80 
4.78 

4.80 
4. 



'8 



7 

r7 

4.75 
4.77 
4.75 
4.75 
4.75 
4.73 
4.73 
4.73 
4.73 

D. 1". 



Cosine. 



9.964026 
.963972 
.963919 
.963865 
.983811 
.963757 
.963704 
.963650 
.963598 
.963542 
.963488 

9.963434 
.963379 
.963325 
.963271 
.963217 
.963163 
.963108 
.963054 
.962999 
.962945 



.962781 
.962727 
.962672 
.962617 
.962562 
.962508 
.962453 



9.962343 



.962233 
.962178 
.962123 
.962067 
.962012 
.961957 
.961902 
.961846 

1.961791 
.961735 
.961680 
.961624 
.961569 
.961513 
.961458 
.961402 
.961346 
.961290 

1.961235 
.961179 
.961123 
.961067 
.961011 



.960843 

.960786 

9.960730 



Sine. 



D.l\ 



.90 
.88 
.90 
.90 
.90 
.88 
.90 
.90 
.90 
.90 
.90 

.92 
.90 
.90 
.90 
.90 
.92 
.90 
.92 
.90 
.92 

.90 
.92 
.90 
.92 
.92 
.92 
.90 



.92 

.92 
.92 
.92 
.92 
.93 
.92 
.92 
.92 
.93 
.92 



.92 
.93 
.92 
.93 
.92 
.93 
.93 
.93 
.92 

.93 
.93 
.93 
.93 
.93 
.93 
.93 
.95 
.93 

D. 1". 

192 



Tang. 



9.627852 



.628554 
.628905 
.629255 
.629606 
.629956 
.630306 
.630656 
.631005 
.631355 

9.631704 
.632053 
.632402 
.632750 
.633099 
.633447 
.633795 
.634143 
.634490 
.634838 

9.635185 
.635532 
.635879 
.636226 
.636572 
.636919 
.637265 
.637611 
.637956 



9.638647 



.639337 
.639682 
.640027 
.640371 
.640716 
.641060 
.641404 
.641747 

9.642091 
.642434 
.642777 
.643120 
.643463 
.643806 
.644148 
.644490 
.644832 
.645174 

9.645516 
.645857 
.616199 
.648540 
.646881 
.647222 
.647562 
.647903 
.648243 

9.648583 

Cotang. 



D. 1", 



5.85 
5.85 
5.85 
5.83 
5.85 
5.83 
5.83 
5.83 
5.82 
5.83 
5.82 

5.82 
5.82 
5.80 
5.82 
5.80 
5.80 
5.80 
5.78 
5.80 
5.78 

5.78 
5.78 
5.78 
5.77 
5.78 
5.77 
5.77 
5.75 
5.77 
5.75 

5.75 
5.75 
5.75 
5.75 
5.73 
5.75 
5.73 
5.73 
5.72 
5.73 

5.72 
5.72 
5.72 
5.72 
5.72 
5.70 
5.70 
5.70 
5.70 
5.70 

5.68 
5.70 
5.68 
5.68 
5.68 
5.67 
5.68 
5.67 
5.67 

D. r. 



Cotang. 



10.372148 
.371797 
.371446 
.371095 
.370745 
.370394 
.370044 
.369694 



.368645 

10.368296 
.367947 
.367598 
.367250 
.366901 
.366553 
.366205 
.365857 
.365510 
.365162 

10.364815 
.364468 
.364121 
.363774 
.363428 
.363081 
.362735 
.362389 
.362044 
.361698 

10.361353 
.361008 
.360663 
.360318 
.359973 
.359C29 
.359281 
.358940 
.358596 
.358253 

10.357909 
.357566 
.357223 
.356880 
.356537 
.356194 
.355852 
.355510 
.355168 
.354826 

10.354484 
.354143 
.353801 
.353460 
.353119 
.352778 
.352438 
.352097 
.351757 

10.351417 

Tang. 



66* 



24' 



COSINES, TANGENTS, AND COTANGENTS. 



155* 



Sine. 



D. 1\ 



9.600313 
.609.597 
.808680 

.610164 
.610447 

.611012 
.611204 
.611576 
.611858 
.612140 

11 I 9.612421 

12 I .61*702 
.612933 
.613264 ' 
.613545 
.613825 

17 I .614105 

18 .614385 : 

19 .614665 i 

20 .614944 





1 
2 
3 

4 
5 
6 
7 
8 
9 
10 






13 
14 
15 
16 



9.G15223 
.615502 
.615781 
.616060 



21 
22 
23 
24 
25 

26 .616616 

27 .616894 

. 617450 

.617727 



31 
32 
33 
34 
So 
36 
37 
38 
39 
40 

41 
42 
43 
44 
45 
46 
47 
43 
49 
50 

51 
52 
53 

54 
55 
56 
57 
58 
59 
60 



9.618004 
.618£81 

.618558 
.618834 

.619110 
.619386 
.619662 



.680213 

.620488 

9.620763 

.621313 
.621587 
.621861 
.6-221:35 
.622409 



.622956 



9.623502 
.623774 
.624047 
.624319 
.624591 



.6251&5 

.625406 

.625677 

9.625948 



4.73 

4.72 
4.73 
4.72 
4.70 
4.72 
4.70 
4.70 
4.70 
4.70 
4.68 

4.68 
4.63 
4.68 
4.68 
4.67 
4.67 
4.67 
4. 07 
4.6.5 
4.65 

4.65 
4.65 
4.65 
4.63 
4.63 
4.63 
4.63 
4.63 
4.62 
4.-J2 

4. 02 
4.02 
4.60 
4.60 
4.60 
4.60 
4.60 
4.53 
4.53 
4.53 

4. 58 
4.53 
4.57 
4.57 
4.57 
4.57 
4.55 
4.57 
4.55 
4.55 

4.53 
4.55 
4.53 
4.53 
4.53 
4.53 
4.52 
4.52 
4.52 



Cosine. 



d. r 



Tang. 



D. r. ! Cotang. 



9 . 960790 
.960674 
.960618 
.960561 
. 960605 
.960448 
.960392 
.960335 
.960279 
.960222 
.960165 

9.960109 
.960052 
.959995 
.959933 
.960882 
.959825 
.950763 
.959711 
.959654 
.959596 

9.959530 

.1534-2 
.959425 

.959310 
.959253 
.959195 
.959138 

.959060 
.959023 

9.958965 



.958850 

.958792 
.958734 
.958677 
.958619 
.958561 
.958503 
.953445 

9.958387 
.05-320 
.953271 
.053213 
.958154 
.958096 
.968088 
.957979 
.957921 
.957363 

9.957804 
.957746 
.957657 
.957623 
.957570 
.957 11 
.957452 
.957393 
. 955335 

9.957276 



.93 
.93 
.95 
.98 

.95 
.93 
.95 
.93 
.95 
.95 
.93 

.95 
.95 
.95 
.93 
.95 ; 
.95 ! 
.95 
.95 i 
.97 ! 
.95 

.95 

.95 ! 

.95 
.97 
.95 
.97 
.95 
.97 
.95 

.95 

.97 

.97 

.97 
.97 

.97 
.97 

.97 

.97 
.97 
.97 
.98 
.97 
.97 
.03 
.97 
.97 
.98 

.97 
.98 
.98 
.97 
.98 
.98 
.98 
.97 
.98 



9.618563 

.649263 

.649602 

.649942 

.650281 

.650620 

.650959 , 

.651297 

.651636 : 

.651974 

9.652312 I 
.652650 ; 
.652933 
.653326 

> .653663 ; 

! .654000 
.654337 
.654674 ; 
.655011 
.655348 

9.6556S4 
.656020 
.656356 
.656692 
.657023 
.657364 
.657699 
.658034 
.658369 
.658704 

9.659039 
.659:373 
.659708 
.660042 
.660376 
.660710 
.661043 
.661377 
.661710 
.662043 

9.662376 
.662709 
.663042 
.663375 
.663707 
.664039 
.664371 
.664703 
.665035 
.665366 

9.665693 
.666029 
.666360 
.666691 
.667021 
.667352 
.667682 
.668013 
.668343 

9.668673 



5.67 
5.67 
5.65 
5.67 
5.65 
5.65 
5.65 
5.63 
5.65 
5.63 
5.63 

5.63 
5.63 
5.63 

5.62 
5.62 
5.62 
5.62 
5.62 
5.62 
5. GO 

5.60 
5.60 
5.60 
5.60 
5.60 
5.58 
5.53 
5.58 
5.58 
5.58 

5.57 
5.58 
5.57 
5.57 
5.57 
5.55 
5.57 
5.55 
5.55 
5.55 

5.55 
5.55 
5.55 
5.53 
5.53 
5.53 
5.53 
6,58 
5.52 
5.53 

5.52 
5.52 
5.52 

5.50 
5.52 
5.50 
5.52 
5. .50 
5.50 



60 
59 
58 
57 
56 
55 
54 
53 
52 



10.351417 
.351077 
.350737 
.350398 
.350058 
.349719 
.349380 
.349041 
.348703 
.348364 51 
.348026 50 

10.347688 I 49 

.347350 43 

.347012 47 

.346674 46 

.346337 45 

.346000 44 

.345663 43 



.345326 
.344989 
.344652 

10.344316 



42 
41 
40 

39 



.343644 
.343308 
.342972 
.342636 
.342301 
.341966 32 
.341631 31 
.341296 ; 30 

10.340961 29 
.340627 
.340292 
.339958 
.330' 
.339290 24 
.338957 23 



37 
36 
&5 
34 
33 



27 
26 
25 



.338623 
.338290 
.337957 

10.337624 
. .337291 



.336625 
.366293 
.335961 
.3356.29 
.335297 
.334965 
.334634 

10.33-J302 
.333971 
.333640 
.333309 
.332979 
.332643 
.332313 
.331987 
.331657 

10.331327 



L_ 



Cosine. D. 1'. 



Sine. D. 1*. , Cotang. D. 1*. Tang. 
193 



65° 



25° 



LOGABITHMIC SIXES, 



154° 



Sine. 



D. 1'. Cosine. D. 1". | Tang. D. 1". Cotang, 




1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

11 

n 

13 
11 
15 
16 
17 
13 
19 
20 

21 

22 
23 
24 
25 
26 
27 
23 
29 
30 

31 
32 
33 
34 

35 
36 
37 
38 
39 
40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



9.625948 
.620219 
.626490 
.626760 
.627030 
.627300 
.627570 
.627840 
.628109 
.628373 
.628647 

9.628916 
.629185 
.629453 
.629721 
.629989 
.630257 
.630524 
.630792 
.631059 
.631326 

9.631593 
.631859 
.632125 
.632392 
.632658 
.632923 
.633189 
.633454 
.633719 
.633984 

9.634249 
.634514 
.634778 
.6:35042 
.635306 
.635570 
.635834 
.636097 
.636360 
.636623 

9.636886 
.637148 
.637411 
.637673 
.637935 
.638197 
.63&458 
.638720 
.638981 
.639242 

9.639503 
.639764 
.640024 
.6402S4 
.640544 
.640804 
.641064 
.641324 
.641583 

9.641842 



Cosine. 



4.52 
4.52 
4.50 
4.50 
4.50 
4.50 
4.50 
4.48 
4.48 
4.48 
4.48 

4.43 
4.47 
4.47 
4.47 
4.47 
4.45 
4.47 
4.45 
4.45 
4.45 

4.43 
4.43 
4.45 
4.43 
4.42 
4.43 
4.42 
4.42 
4.42 
4.42 

4.42 

4.40 
4.40 
4.40 
4.40 
4.40 
4.38 
4 38 
4.38 
4.38 

4.37 
4.38 
4.37 
4.37 
4.37 
4.35 
4.37 
4.35 
4.35 
4.35 

4.35 
4.33 
4.33 
4.33 
4.33 
4.33 
4.33 
4.32 
4.32 



D. 1\ 



9.957276 
.95721/ 

.957158 
.957099 
.957040 
.956981 
.956921 
.956862 
.956803 
.956744 
.956684 

9.956625 
.956566 
.956506 
.956447 
.956387 
.956327 
.956268 
.956208 
.956148 
.956089 

9.956029 
.955969 
.955909 
.955849 
.955789 
.955729 
.955669 
.955609 
.955.543 
.955488 

9.955428 
.955368 
.955307 
.955247 
.955186 
.955126 
.955065 
.955005 
.954944 
.954883 

9.954823 
.954762 
.9.54701 
.954640 
.9.54579 
.954518 
.954457 
.954396 
.954335 
.954274 

9.954213 
.954152 
.954090 
.954029 
.953968 
.953906 
.953845 
.953783 
.9.53722 

9.953660 



Sine. 



.98 

.98 

.98 

1.00 



1.00 
.98 

.93 
1.00 

.98 
1.00 
1.00 

.98 
1.00 
1.00 

.98 
1.00 

1.00 
1.00 
1.00 
1.00 
1.00 
1.00 
1.00 
.93 
1.00 
1.00 

1.00 

1.02 
1.00 
1.02 
1.00 
1.02 
1.00 
1.02 
1.02 
1.00 

1.02 
1.02 
1.02 
1.02 
1.02 
1.02 
1.02 
1.02 
1.02 
1.02 

1.02 
1.03 
1.02 
1.02 
1.03 
1.02 
1.03 
1.02 
1.03 



9.668673 
.669002 
.669332 
.669661 
.669991 
.670320 
.670649 
.670977 
.671306 
.671635 
.671963 

9.672291 
.672619 
.672947 
.673274 
.673602 
.673929 
.674257 
.674584 
.674911 
.675237 

9.675564 
.675890 
.676217 
.676543 
.676869 
.677194 
.677520 
.677846 
.678171 
.678496 

9.678821 
.679146 
.679471 
.679795 
.680120 
.680444 



.681092 
.681416 
.681740 



.682387 
.682710 



.683356 
.683679 
.684001 
.684324 
.684646 
.684968 

9.685290 
.685612 
.685934 
.686255 
.686577 



.687219 

.687540 

.687861 

9.688182 



115* 



D. 1\ I Cotang. 
194 



5.48 
5.50 
5.48 
5.50 
5.48 
5.48 
5.47 
5.48 
5.48 
5.47 
5.47 

5.47 
5.47 
5.45 
5.47 
5.45 
5.47 
5.45 
5.45 
5.43 
5.45 

5.43 
5.45 
5.43 
5.43 
5.42 
5.43 
5.43 
5.42 
5.42 
5.42 

5.42 
5.42 
5.40 
5.42 
5.40 
5.40 
5.40 
5.40 
5.40 
5.38 

5.40 
5.38 
5.38 
5.38 
5.38 
5.37 
5.38 
5.37 
5.37 
5.37 

5.37 
5.37 
5.35 
5.37 
5.35 
5.35 
5.35 
5.35 
5.35 



10.331327 

.330998 
.330668 
.330339 
.330C09 
.329680 
.329851 
.329023 
.328694 
.328365 
.328037 

10.327709 
.327381 
.327053 
.326726 
.326398 
.326071 
.325743 
.325416 
.325089 
.324763 

10.324436 
.324110 
.323783 
.323457 
.323131 
.322806 
.322480 
.322154 
.321829 
.321504 

10.321179 
.320854 
.320529 
.320205 
.319880 
.319556 
.319232 
.318908 
.318584 
.318260 

10.317937 
.317613 
.317290 
.316967 
.316644 
.316321 
.315999 
.315676 
.315354 
.315032 

10.314710 
.314388 
.3140C6 
.313745 
.313423 
.313102 
.312781 
.312460 
.312139 

10.311818 



D. r. I Tang. 



64* 



428 COSINES, TANGENTS, AND COTANGENTS. ^^ 



' Sine. D. 1*. i Cosine. D. 1'. j Tang. D. 1'. Cotang. 



9. £25511 
.825051 
.825791 
.825931 
.826071 
.820211 
.826351 
.826491 
.826631 
.826770 
.826910 

9.827049 

.827189 
.827323 
.827407 
.827603 
.827745 
.827884 
.82802-3 
.828162 
.828301 

9.828439 
.828573 
.828716 
.828855 



.829131 
.829269 
.829407 
.829545 
.829683 

9.829821 
.829959 
.830097 
.830234 
.830372 



.830646 
.830784 
.830921 
.831058 

9.831195 
.831332 
.831469 
.831606 
.831742 
.831879 
.832015 
.832152 
.832288 
.832425 

9-832501 
.£32697 
.832833 
.832969 
.833105 
.833241 
.833377 
.833512 
.833648 

9.833783 ; 



2. S3 
2.33 
2.33 
2.33 
2.a3 
2.33 
2.33 
2.33 
2.32 
2.33 
2.32 

2. S3 
2.32 
2.32 
2.32 
2.32 
2.32 
2.32 
2.32 
2.32 
2.30 

2.32 
2.30 
2.32 
2.30 
2.30 
2.30 
2.30 
2.30 
2.30 
2.30 

2.30 
2.30 
2.28 
2.30 
2.28 
2.28 
2.30 
2.28 
2.28 
2.28 

2.28 
2.28 
2.28 
2.27 
2.28 
2.27 
2.28 
2.27 
2.28 
2.27 

2.27 
2.27 
2.27 
2.27 
2.27 
2.27 
2.25 
2.27 
2.25 



' I Cosine. I D. 1', 



9.871073 

.870960 
.870846 
.870732 
.870618 
.870504 
.870390 
.870276 
.870101 
.870047 
.869933 

9.869818 
.869704 
.869569 
.869474 
.869360 
.869245 
.869130 
.869015 
.868900 
.868785 

9.868670 
.868555 
.668440 
.868324 
.868209 
.668093 
.667978 
.867602 
.867747 
.867631 

9.8-67515 
.867399 
.867283 
.867167 
.867051 
.866935 
.866819 
.866703 



.866470 : 

9.866353 
.866237 ! 

.866120 
.866004 
.665887 
.865770 
.865653 
.865536 
.865419 
.865302 

9.865185 
.865068 
.864950 
.864833 
.864716 
.864598 
.864481 
.864363 
.864245 

9.864127 



1.88 
1.90 
1.90 
1.90 
1.90 
1.90 
1.90 
1.92 
1.90 
1.90 
1.92 

1.90 
1.92 
1.92 
1.90 
1.92 
1.92 
1.92 
1.92 
1.92 
1.92 



1.93 
1.93 
1.93 
1.93 
1.93 
1.93 
1.93 
1.95 
1.93 
1.95 

1.93 

1.95 
1.93 
1.95 
1.95 
1.95 
1.95 
1.95 
1.95 
1.95 

1.95 

1.97 
1.95 
1.95 
1.97 
1.95 
1.97 
1.97 
1.97 



9.054437 
.954691 
.954946 
.955200 
.955454 
.955708 
.955961 
.956215 
.956409 
.956703 
.956977 

9.957231 
.957465 
.G577S9 
.957993 
.958247 
.958500 
.958754 
.959008 
.959262 
.959516 

9.959709 
.960023 
.960277 
.960530 
.960784 
.961038 
.961292 
.961545 
.961799 
.962052 

9.962306 
.96256-0 
.962813 
.963067 
.963320 
.963574 
.963828 
.964081 
.964335 
.964588 

9.964842 
.965095 
.965349 
.965602 
.965855 
.966109 



.966616 
.966869 
.967123 

9.967376 
.967629 
.967883 
.968136 



.968643 



.969149 

.969403 

9.969656 



W 



Sine. I D. 1". II Cotang. 
105 



4.C3 
4.25 
4.23 
4.23 
4.23 
4.22 
4.23 
4.23 
4.23 
4.23 
4.23 

4.23 
4.23 
4.23 
4.23 
4.22 
4.23 
4.23 
4.23 
4.23 
4.22 

4.23 
4.23 
4.22 
4.23 
4.23 
4.23 
4.22 
4.23 
4.22 
4.23 

4.23 
4.22 
4.23 
4.22 
4.23 
4.23 
4.22 
4.23 
4.22 
4.23 

4.22 
4.23 
4.22 
4.22 
4.23 
4.22 
4.23 
4.22 
4.23 
4.22 

4.22 
4.23 
4.22 
4.22 
4.23 
4.22 
4.22 
4.23 
4.22 



10.045553 
.045309 
.045054 
.044800 
.044546 
.044292 
.044039 
.043785 
.043531 
.043277 
.043023 

10.042769 
.042.515 
.042261 
.042007 
.041753 
.041500 
.041246 
.040992 
.0407 8 
.040484 

10.040231 
.039977 
.039723 
.039470 
.039216 



.038708 
.038455 
.038201 
.037948 

10.037694 
.037440 
.037187 
.036933 
.036680 
.036426 
.036172 
.035919 
.035665 
.035412 

10.035158 
.034905 
.034651 



.034145 
.033891 
.0330.38 
.033384 
.033131 
.032877 

10.032624 
.032371 
.032117 
.031864 
.031611 
.031357 
.031104 
.030851 
.030597 

10.030344 



D. 1'. i Tang. 



47* 



43° 



LOGARITHMIC SIXES, 



136° 



Sine. D. 1". Cosine. D. 1\ Tang. D. 1". Cotang. ' 






9.833783 


1 


.833919 


2 


.834054 


3 


.834189 


4 


.834325 


5 


.834460 


6 


.834595 


7 


.834730 


8 


.834865 


9 


.834999 


10 


.835134 


11 


9.8352C9 


12 


.835403 


n 


.835538 


14 


.835672 


15 


.835807 


16 


.835941 


17 


.836075 


18 


.836209 


19 


.836343 


20 


.836477 


21 


9.836611 


22 


.836745 


23 


.836878 


24 


.837012 


25 


.837146 


26 


.837279 


27 


.837412 


28 


.837546 


29 


.837679 


30 


.837812 


31 


9.837945 


32 


.838078 


33 


.838211 


34 


.838344 


35 


.838477 


36 


.838610 


37 


.838742 


38 


.838875 


39 


.839007 


40 


.839140 


41 


9.839272 


42 


.839404 


43 


.839536 


44 


.839668 


45 


.839800 


46 


.839932 


47 


.840064 


48 


.840196 


49 


.840328 


50 


.840459 


51 


9.840591 


52 


.840722 


53 


.840854 


54 


,840985 


55 


.841116 


56 


.841247 


57 


.841378 


58 


.841509 


59 


.841640 


60 


9.841771 



Cosine. 



2.27 
2.25 
2.25 
2.27 
2.25 
2.25 
2.25 
2.25 
2.23 
2.25 
2.25 

2.23 
2.25 
2.23 
2.25 
2.23 
2.23 
2.23 
2.23 
2.23 
2.23 

2.23 
2.22 
2.23 
2.23 
2.22 
2.22 
2.23 
2.22 
2.22 
2.22 

2.22 
2.22 
2.22 
2.22 
2.22 
2.20 
2.22 
2.20 
2.22 
2.20 

2.20 
2.20 
2.20 
2.20 
2.20 
2.20 
2.20 
2.20 
2.18 
2.20 

2.18 
2.20 
2.18 
2.18 
2.18 
2.18 
2.18 
2.18 
2.18 



9.864127 
.864010 
.863892 
.863774 
.863656 
.863538 
.863419 



D. r. 



.863004 
.862946 

9.862827 
.862709 
.862590 
.862471 



.862234 
.862115 



.861877 
.861758 

9.861638 
.861519 
.861400 
.861280 
.861161 
.861041 
.860922 



.860682 
.860562 

9.860442 
.860322 



.660082 



.859842 
.859721 
.859601 
.859480 
.859360 

9.859239 
.859119 
.858998 
.858877 
.858756 
.858635 
.858514 



.858272 
.858151 

9.858029 
.857908 
.857786 
.857665 
.857543 
.857422 
.857300 
.657178 
.857056 

9.856934 



1.95 

1.97 
1.97 
1.97 
1.97 
1.98 
1.97 
1.97 
1.98 
1.97 
1.98 

1.97 
1.98 
1.98 
1.97 
1.98 
1.98 
1.98 
1.98 
1.98 
2.00 

1.98 
1.98 
2.00 
1.98 
2.00 
1.98 
2.00 
2.00 
2.00 
2.00 

2.00 
2.00 
2.00 
2.00 
2.00 
2.02 
2.00 
2.02 
2.00 
2.02 

2.00 
2.02 
2.02 
2.02 
2.02 
2.02 
2.02 
2.02 
2.02 
2.03 

2.02 
2.03 
2.02 
2.03 
2.02 
2.03 
2.03 
2.03 
2.03 



133° 



Sine. ! D. 1'. I 
196 



9.969656 
.969909 
.970162 
.970416 
.970669 
.970922 
.971175 
.971429 
.971682 
.971935 
.972188 

9.972441 

.972695 
.972948 
.973201 
.973454 
.973707 
.973960 
.974213 
.974466 
.974720 

9.974973 
.975226 
.975479 
.975732 
.975985 
.976238 
.976491 
.976744 
.976997 
.977250 

T503 
,y<<756 
.978009 
.978262 
.978515 
.978768 
.979021 
.979274 
.979527 
.979780 

9.980033 
.980286 
.980538 
.980791 
.981044 
.981297 
.981550 



9.97 



.982056 



9.982562 
.982814 
.983067 



.983573 



.984079 

.984332 

.984584 

9.984837 



4.22 
4.22 
4.23 
4.22 
4.22 
4.22 
4.23 
4.22 
4.22 
4.22 
4.22 

4.23 
4.22 

4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.23 
4.22 

4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 

4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 

4.22 
4.20 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 

4.20 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.20 
4.22 



10.0S0344 
.080091 



.029584 
.029331 
.029078 
.028825 
.028571 
.028318 
.028065 
.027812 

10.027559 
.027805 
.027052 
.026799 
.026546 
.026293 
.026040 
.025787 
.025534 
.025280 

10.025027 
.024774 
.024521 
.024268 
.024015 
.023762 
.023509 
.023256 
.023003 
.022750 

10.022497 
.022244 
.021991 
.021738 
.021485 
.021232 
.020979 
.020726 
.020473 
.020220 

10.019967 
.019714 
.019462 
.019209 
.018956 
.018703 
.018450 
.018197 
.017944 
.017691 

10.017438 
.017186 
.016933 
.016680 
.016427 
.016174 
.015921 
.015668 
.015416 

10.015163 



Cotang. I D. 1". I Tang. 



46° 



44° 



COSINES, TANGENTS, AND COTANGENTS. 



135« 



' Sine. D. 1". Cosine. D. 1". Tang. D. 1". Cotang. ' 




1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

!Z 

134° 



9.841771 
.841902 
.842033 
.842163 
.842294 
.842424 
.842555 
.842685 
.842815 
.842946 
.843076 

9.843206 
.843336 
.843466 
.843595 
.843725 
.843855 
.843984 
.844114 
.844243 
.844372 

9.844502 
.844631 
.844760 
.844889 
.845018 
.845147 
.845276 
.845405 
.845533 
.845662 

9.845790 
.845919 
.846047 
.846175 
.846304 
.846432 
.846560 



.846816 
.846944 

9.847071 
.847199 
.847327 
.847454 

.847582 
.847709 
.847836 
.847964 
.848091 
.848218 

9.848345 

.848472 
.848599 
.848726 
.848852 
.848979 
.849106 
.849232 
.849359 
9.849485 



Cosine. 



2.18 
2.18 
2.17 
2.18 
2.17 
2.18 
2.17 
2.17 
2.18 
2.17 
2.17 

2.17 
2.17 
2.15 
2.17 
2.17 
2.15 
2.17 
2.15 
2.15 
2.17 

2.15 
2.15 
2.15 
2.15 
2.15 
2.15 
2.15 
2.13 
2.15 
2.13 

2.15 
2.13 
2.13 
2.15 
2.13 
2.13 
2.13 
2.13 
2.13 
2.12 

2.13 
2.13 
2.12 
2.13 
2.12 
2.12 
2.13 
2.12 
2.12 
2.12 

2.12 
2.12 
2.12 
2.10 
2.12 
2.12 
2.10 
2.12 
2.10 



D. r 



9.856934 
.856812 
.856690 
.856568 
.856446 
.856323 
.856201 
.850078 
.855956 
.855833 
.855711 

9.855583 
.855465 
.855342 
.855219 
.855096 
.854973 
.854850 
.854727 
.854603 
.854480 

9.854356 
.854233 
.854103 

.853986 



.853738 
.853614 
.853490 



.853242 

9.853118 
.852994 
.852869 
.852745 
.852620 
.852496 
.852371 
.852247 
.852122 
.851997 

9.851872 
.851747 
.851622 
.851497 
.851372 
.851246 
.851121 
.850996 
.850870 
.850745 

9.a50619 
.850493 
.850368 
.850242 
.850116 
.849990 
.849864 
.8497:38 
.849611 

9.849485 



Sine. 



2.03 
2.03 
2.03 
2.03 
2.05 
2.03 
2.05 
2.03 
2.05 
2.03 
2.05 

2.05 
2.05 
2.05 
2.05 
2.05 
2.05 
2.05 
2.07 
2.05 
2.07 

2.05 
2.07 
2.05 
2.07 
2.07 
2.07 
2.07 
2.07 
2.07 
2.07 

2.07 
2.08 
2.07 
2.08 
2.07 
2.08 
2.07 
2.08 
2.08 
2.08 

2.08 
2.08 
2.08 
2.08 
2.10 
2.08 
2.08 
2.10 
2.08 
2.10 

2.10 
2.08 
2.10 
2.10 
2.10 
2.10 
2.10 
2.12 
2.10 



9. 984837 
.985090 
.985343 
.985596 



.986101 
.980554 
.986607 
.986860 
.987112 
.987365 

9.987018 
.987871 
.988123 
.988376 



.989134 
.989387 
.989640 



9.990145 
.990398 
.990651 
.990903 
.991156 
.991409 
.991662 
.991914 
.992167 
.992420 

9.992672 
.992025 
.993178 
.99&431 
.993C83 
.993936 
.994189 
.994441 
.994694 
.994947 

9.995199 
.995452 
.995705 
.995957 
.996210 
.996463 
.996715 



d. r. 



197 



.997221 
.997473 

9.997726 
.997979 
.998231 

.998484 
.998737 
.998989 
.999242 
.999495 
.999747 
10.000000 



4.22 

4.22 
4.22 
4.20 
4.22 

4.22 
4.22 
4.22 
4.20 
4.22 
4.22 

4.22 
4.20 

4.22 
4.22 
4.22 
4.20 
4.22 
4.22 
4.22 
4.20 

4.22 
4.22 
4.20 
4.22 
4.22 
4.22 
4.20 
4.22 
4.22 
4.20 

4.22 
4.22 

4.22 
4.20 
4.22 
4.22 
4.20 
4.22 
4.22 
4.20 

4.22 
4.22 
4.20 
4.22 
4.22 
4.20 
4.22 
4.22 
4.20 
4.22 

4.22 
4.20 
4.22 
4.22 
4.20 
4.22 
4.22 
4.20 
4.22 



Cotang. I D. 1*. 



10.015163 
.014910 
.014657 
.014404 
.014152 
.013899 
.013646 
.013393 
.013140 
.012888 
.012635 

10.012382 
.012129 
.011877 
.011624 
.011371 
.011118 
.010866 
.010613 
.010360 
.010107 

10.009855 
.009602 
.009349 
.009097 
.008S44 
.008591 
.008338 
.008086 
.007833 
.007580 

10.007328 
.007075 
.006822 
.006569 
.006317 
.006064 
.005811 
.005559 
.005306 
.005053 

10.004801 
.004548 
.004295 
.004043 
.003790 
.003537 
.003285 
.003032 
.002779 
.002527 

10.002274 
.002021 
.001769 
.001516 
.001263 
.001011 
.000758 
.000505 
.000253 

10.000000 



Tang. 



45* 



NATURAL FUNCTIONS. 



NATURAL SINES AND COSINES. 



1 

"o 


0° 


1° 


2° 


3° 


4° 


/ 
60 


Sine 


Cosin 


Sine 
.01745 


Cosin 


Sine 
.03490 


Cosin 


Sine 


Cosin 


Sine | Cosin 


.00000 


One. 


.99985 


.99939 


.05234 


. 99863 ! .06976 .99756 


1 


.10029 


One. 


.01774 


.99984 


.03519 


.99938 


.05263 


.99861 .07005 .99754 


59 


2 


.00058 


One. 


.01803 


.99984 


.03548 


.99937 


.05292 


.99860 .07034 .99752 


58 


3 


.00087 


One. 


.01832 


.99983 


'.03577 


.99936 


.05321 


.99858 ! .0706-3 .99750 


57 


4 


.00116 


One. ; 


.01862 


.99983!. 03606 


.99935 


.05350 


.99857 .07092.99748 


56 


5 


.00145 


One. 


.01891 


.99982! 1.03635 


.99934 


.05379 


.99855, 


.07121 .99746 


55 


6 


.00175 


One. ! 


.01920 


.99982:!. 03664 


.99933 


.05408 


.998541 


.07150 .99744 


54 


7 


.00204 


One. i 


.01949 


.99981 .03693 


.99932 


.05437 


.99852| 


.07179 .99742 


53 


8 


.00233 


One. | 


.01978 


.99980 .03723 


.99931 


.05466 


.99851 


.0720S .99740 


52 


9 


.00262 


One. 


.02007 


.99980 .03752 


.99930 


.05495 


.99849 .07237.99738 


51 


10 


.00291 


One. 


.02036 


.99979 .03781 


.99929 


.05524 


.99847, 


.07266 


.99736 


50 


11 


.00320 


.99999 


.02065 


.99979 1 .03810 


.99927 


.05553 


.99846 


.07295 


.99734 


49 


12 


.00349 


.99999 


.02094 


. 99978 j i.03839 


.99926 


.05582 


.99844 


.07324 


.99731 


48 


13 


.00378 


.99999 


.02123 


.99977!:. 03868 


.99925 


.05611 


.99&42 


.07353 


.99729 


47 


14 


.00407 


.99999 


.02152 


.99977 ! .03897 


.99924 


.05640 


.99841, 


.07382 


.99727 


46 


15 


.00436 


.99999 


.02181 


.99976 .03926 


.99923 


.05669 


.998391 


.07411 


.99725 


45 


16 


.00465 


.99999 


.02211 


.99976 1 1 . 03955 


.99922 


.05698 


.99838 ! 


.07440 


.99723 


44 


17 


.00495 


.99999: 


.02240 


.999751 .03984 


.99921 


.05727 


.99836! 


.07469 


.99721 


43 


18 


.00524 


.99999, 


.02269 


.99974 .04013 


.99919 


.05756 


.99834! 


.07498 


.99719 


42 


19 


.00553 


.99998 


.02298 


.99974 .04042 


.99918 


.05785 


.99833; 


.07527 


.99716 


41 


20 


.00582 


.99998 


.02327 


.99973 i. 04071 
.99972 !' .04100 


.99917 


.06814 


.998311 


.07556 


.99714 


40 


21 


.00611 


.99998 


.02356 


.99916 


.05S44 


.99829! 


.07585 


.99712 


39 


22 


.00640 


.99998 


.02385 


.99072 .04129 


.99915 


! .05873 


.99827, 


.07614 


.99710 


38 


23 


.00669 


.99998; 


.02414 


.99971 .04159 


.99913 


.05902 


.99826' 


,<>7643 


.99708 


37 


24 


.00698 


.99998 


.02443 


.99970J .04188 


.99912 


! .05931 


.99824 ; 


.07672 


.99705 


36 


25 


.00727 


.99997 


.02472 


.99969; .04217 


.99911 


] .05960 


.99822 


.07701 


.99703 


35 


26 


.00756 


.99997 


.02501 


.99969 ' .04246 


.99910 


: .05989 


.99821 


.07730 


.99701 


34 


27 


.00785 


.99997 


.02530 


.99968; .04275 


.99909 


1 .06018 


.99819 


.07759 


.99699 


33 


28 


.00814 


.99997 


.02560 


.999671. 04304 


.99907 


.06047 


.99817; 


.07788 


.99696 


32 


29 


.00844 


.99996 


.02589 


.99906 .04333 


.99906 


.00076 


.99815 


.07817 


.99694 


31 


30 


.00873 


.99996 


.02618 


.99966 


, .04362 


.99905 


j .06105 


.99813 


.07846 


.99692 


30 


31 


.00902 


.99996 


.02647 


.99965 


1.04391 


.99904 


1.06134 


.99812 ' 


.07875 


.99689 


29 


32 


.00931 


.99996 


.02676 


.99964 


.04420 


.99902 


.C0163 


.99810 .07904 


.99687 


28 


33 


.00960 


.99995 


.02705 


.99963 


.04449 


.99901 


.06192 


.99808 


.07933 


.99685 


27 


34 


.00989 


.99995 


.02734 


.99963 


1.04478 


.99900 


.06221 


.99806 


.07962 


.99683 


26 


35 


.01018 


.99995 


.02763 


.99962 I .04507 


.99898 


.06250 


.99804 


.07991 


.99680 


25 


36 


.01047 


.99995 


.02792 


.99961 


! .04536 


.99897 


j .06279 


.99803 


.08020 


.99678 


24 


37 


.01076 


.99994 


.02821 


.99900 


-.04565 


.99896 


I.OCS08 


.99801 


.08049 


.99676 


23 


38 


.01105 


.99994 


.02850 


.99959 


.04594 


.99894 


! .06337 


.99799 


.08078 


.99673 


22 


39 


.01134 


.99994 


.02879 


.99959 


.04023 


.99893 


! .06366 


.99797 


.08107 


.99671 


21 


40 


.01164 


.99993 


.02908 


.99958! .04653 


.99892 


.06395 


.99795 


.08136 


.99668 


20 


41 


.01193 


.99993 


.02938 


.99957 '.046S2 


.99890 


.06424 


.99793 ' 


.08165 


.99666 


19 


42 


.01222 


.99993 


.02967 


.99956 .04711 


.99889 


.06453 


.99792 


.08194 


.99664 


18 


43 


.01251 


.99992 


.02996 


.99955 .04740 


'.99888 


.06482 


.99790 


.08223 


.99661 


17 


44 


.01280 


.99992 


.03025 


.99954 .04769 


1.99886 


.06511 


.99788 


.08252 


.99659 


16 


45 


.01309 


.99991 


.03054 


.99953 .04798 


.99885 


.06540 


.99786 


.08281 


.99657 


15 


46 


.0ia38 


.99991 


.03083 


. 99952 i .04827 


.99883 


.06569 


.997&4 


.08310 


.99654 


14 


47 


.01367 


.99991 


.03112 


.99952 .04856 


.99882 


•■ .06598 


.99782 


.08339 


.99652 


13 


48 


.01396 


.99990 


.03141 


.99951 .04885 


.99881 


.06627 


.99780 


.08368 


.99649 


12 


49 


.01425 


.99990 


.03170 


.99950 .04914 


.99879 


.06656 


.99778 


.08397 


.99647 


11 


50 


.01454 


.99989 


.03199 


.99949 .04943 


.99878 


.06685 


.99776 


.0&426 .99644 


10 


51 


.01483 


.99989 


.03228 


. 99948 |! . 04972 


.99876 


' .06714 


.99774 


.08455 .99642 


9 


52 


.01513 


.99989 


.03257 


.99947 .05001 


.99875 


.06743 .99772 


.08484 .99639 


8 


53 


.01542 


.99988 


.03286 


.99946 .05030 


.99873 


: .06773 .99770 


.08513 .99637 


7 


54 


.01571 


.99988 


.03316 


.99945 .05059 


.99872 


.06802 ,.99768 


.08542 


.99635 


6 


55 


.01600 


.99987 


.03345 


.99944 .05088 


.99870 


.06831 '.99766 


.08571 


.99632 


5 


56 


.01629 


.99987 


.03374 


.99943 1.05117 


.99869 


i .06860 .99764 


.08600 


.99630 


4 


57 


.01658 


.99986 


.03403 


.99942 .05146 


.99867 


! .06889 .99762 


.08629 .99627 


3 


58 


.01687 


.99986 


.03432 


.99941 .05175 


.99866 


1 .06918 .99760 


.08658 .99625 


2 


59 


.01716 


.99985 


.03461 


.99940 .05205 


.99864 


.06947 .99758 


.08687 .99622 


1 


60 

/ 


.01745 


.99985 


.03490 
Cosin 


.99939 .05234 


.99863 


.06976 .99756 
Cosin | Sine ; 


.08716 .99619 




/ 


Cosin 


Sine 


Sine 


Cosin 


Sine 


Cosin j Sine 


89° 


88° 


87° 


8( 


*• 


85° 



201 



NATURAL SINES AND COSINES. 



/ 

~o 


5° 


6° 


7° 


8« 


9° 


60 


Sine 


Cosin 


Sine 


Cosin 


Sine 

.12187 


Cosin ! 
.99255 


Sine 


Cosin 


Sine 
.15043 


Cosin 
.98709 


.08716 


.99619 


.10453 


.99452 


.13917 


.99027 


1 


.08745 


.99617 


.10482 


.99449 


.12216! 


.99251 


.13946! 


.99023 


.15072 


.98764 


59 


2 


.08774 


.99614 


.10511 


.99446 


. 12245 


.99248 


. 13975 


.99019 


.15701 


.98760 


58 


3 


.0S803 


.99612 


.10540 


.99443 


. 12274 ' 


.99244 


.14004 


.99015 


.15730 


.98755 


57 


4 


.08831 


.99609! 


.10569 


.99440 


.12302 


.99240 


.14033 


.99011 


.15758 


.98751 


56 


5 


.08860 


.99607J 


.10597 


.99437 


.12331 


.99237 


.14061 


.99000 


.15787 


.98746 


55 


6 


.08889 


.99604 


.10626 


.99434-: 


.12300 


.99233 


.14090 


.99002 


.1.5816 


.98741 


54 


7 


.08918 


.99602 


.10655 


.99431 


.12389 


.99230 


.14119 


.98998 


.1.5845 


.98737 


53 


8 


.08947 


.99599 


.10684 


.99428 


.12418 


.99226 


.14148 


.98994 


.15873 


.98732 


52 


9 


.08976 


.99596 


.10713 


.99424 


.12447 


.99222 


.14177 


.98990 


.15902 


.98728 


51 


10 


.09005 


.99594 


.10742 


.99421, 


.12476 


.99219 


.14205 


.98986 


.15931 


.98723 


50 


11 


.09034 


.99591 


.10771 


.99418 


.12504 ' 


. 99215 ' 


.14234 


.98982 


.15959 


.98718 


49 


12 


.09063 


.99588 


.10800 


.99415' 


.12533 


.99211! 


.14263 


.98978 


.15988 


.98714 


48 


13 


.09092 


.99586 


.10829 


.99412 


.12562 


.99208 


.14292 


.98973 


.16017 


.98709 


47 


14 


.09121 


.99583 


.10858 


.99409 


.12591 


.99204 


.14320 


.98969 


.16046 


.98704 


46 


15 


.09150 


.99580 


.10387 


.99406 


.12620 


.99200 


.14349 


.98965 


1.16074 


.98700 


45 


16 


.09179 


.99578 


.10916 


.99402; 


.12049 


.99197 


.14378 


.98961 


.16103 


.98695 


44 


17 


.09208 


.99575 


.10945 


.99399, 


.12078 


.99193 


.14407 


.98957 


.16132 


.98090 


43 


18 


.09237 


.99572 


.10973 


.99396 


.12706 


.99189 


.14436 


.98953 


.16160 


.98686 


42 


19 


.09206 


.99570 


.11003 


.99393 


.12735 


.99186 


.14464 


.98948 


.16189 


.98681 


41 


20 


.09295 


.99567 


.11031 


.99390: 


.12704 


.99182 


.14493 


.98944 


.10218 


.98676 


40 


21 


.09324 


.99564 


.11060 


.99386 


.12793 


.99178 ! 


.14522 


.98940 


.16246 


.98671 


39 


22 


.09353 


.99562 


.11039 


.99333 


.12822 


.99175 


.14551 


.98936! 


.16275 


.98667 


38 


23 


.09382 


.99559 


.11118 


.99380 


.12351 


.99171 ! 


.14580 


.98931 


.16304 


.98662 


37 


24 


.09411 


.99556 


.11147 


.99377 


.12380 


.99167 


.14608 


.98927 


.16333 


.98657 


36 


25 


.09440 


.99553 


.11176 


.99374 


.12908 


.99103 


.14637 


.98923 


.16361 


.98652 


35 


26 


.09469 


.99551 


.11205 


.99370 


.12937 


.99100; 


.14666 


.98919 


.16390 


.98048 


34 


27 


.09498 


.99548 


.11234 


.99367 


.12906 


.99156 


.14695 


.98914 


.16419 


.98043 


33 


28 


.09527 


.99545 


.11263 


.99364 


.12995 


.99152 


.14723 


.98910 


.16447 


.98638 


32 


29 


.09556 


.99542 


.11291 


.99330 


.13024 


.99148, 


.14752 


.98906 


.16476 


.98633 


31 


30 


.09585 


.99540 


.11320 


.99357 | 


.13053 


.99144 


.14781 


.98902 


.16505 


.98629 


30 


31 


.09614 


.99537 


.11349 


.99354 


.13031 


.99141 


.14810 


.98897 


'.16533 


.98624 


29 


32 


.01)642 


.99534 


.11378 


.99351 


.13110 


.991371 


.14838 


.98893 


.16562 


.98619 


28 


33 


.09071 


.99531 


.11407 


.99347 


.13139 


.99133! 


.14867 


.98889 


.16591 


.98614 


27 


34 


.09700 


.99528 


.11436 


.99344 


.13163 


.99129 


.14896 


.98884 


.16620 


.9S609 


26 


35 


.09729 


.99526 


.11465 


.99341 


.13197 


.99125 


.14925 


.98880 


.16648 


.98604 


25 


36 


.09758 


.99523 


.11494 


.99337 


.13223 


.99122 


.14954 


.98876 


.16677 


.98600 


24 


37 


.09787 


.99520 


.11523 


.99334 


.13254 


.99118 


.14982 


.98871 


.16706 


.98595 


23 


38 


.09816 


.99517 


.11552 


.99331 


.13283 


.99114' 


.15011 


.98867 


.10734 


.98590 


22 


39 


.09845 


.99514 


.11580 


.99327 


.13312 


.99110 


.15040 


.98863 


.10763 


.98585 


21 


40 


.09874 


.99511 


.11609 


.99324 


.13341 


.99106 ( 


.15069 


.98858 


, .10792 


.98580 


20 


41 


.09903 


.99508 


.11638 


.99320 ' 


.18370 


.99102 ' 


.15097 


.98854 


1 .16820 


.98575 


19 


42 


.09932 


.99506 


.11667 


.99317 


.13399 


.99098 


.15126 


.98849 


.16849 


.98570 


18 


43 


.09961 


.99503 


.11696 


.99314 


.13427 


.99094 


.15155 


.98845 


.16878 


.98565 


g 


44 


.09990 


.99500 


.11725 


.99310 


.13456 


.99091 


.15184 


.98841 


.10906 


.98561 


16 


45 


.10019 


.99497 


.11754 


•99307 


.13485 


.99087 


.15212 


.98836 


.16935 


.98556 


15 


46 


.10048 


.99494 


1.11783 


.99303 


.13514 


.99083 


.15241 


1.98832 


.16964 


.98551 


14 


47 


.10077 


.99491 


.11812 


.99300 j 


.13543 


.99079 


.15270 


1.98827 


.16992 


.98546 


13 


48 


.10106 


.99488 


.11840 


.99297! 


.13572 


.99075 


.15299 


.98823 


.17021 


.98541 


12 


49 


.10135 


.99485 


.11869 


.99293: 


.13000 


.99071 


.15327 


1.98818 


.17050 


.98536 


11 


50 


.10164 


.99482 


.11898 


. 99290 i 


.13029 


.99067 


.15356 


j .88814 


.17078 


.98531 


10 


51 


.10192 


.99479 


.11927 


.99286 


.13658 


.90003 


.15385 


1.98809 


'.17107 


.98526 


9 


52 


.10221 


.99470 


.11956 


.99283; 


.13037 


.99059 


.15414 


i. 98805 


.17136 


.98521 


8 


53 


.10250 


.99473 


! .11985 


. 99279 '■ 


.13710 


.99055 


.15442 


'.988C0 


.17164 


.98516 


7 


54 


.10279 


.99470 


1.12014 


.99276 


.13744 


.99051 


.15471 


!. 98796 


i .17193 


.98511 


6 


55 


.10308 


.99467 


.12043 


.99272 


.13773 


.99047 


.15500 


1.98791 


.17288 


.98506 


5 


56 


.10337 


.99404 


.12071 


.99269 


.13802 


.99043 


.15529 


.98787 


,.17250 


.98501 


4 


57 


10366 


.99401 


.12100 


.99265 


.13831 


.99039 


.15557 


.98782 


j. 17279 


.98496 


3 


58 


.10395 


.99458 


.12129 


.99262 


.13860 


.99035 


.15586 


.98778 


.17308 


.98491 


2 


59 


.10424 


.99455 


: .12158 


.99258 


.13889 


.99031 


.15615 


.98773 


1.17336 


i. 98486 


1 


60 
/ 


.10453 


.99452 


S .12187 
Cosin 


.99255 
Sine 


.13917 
Cosin 


.99027 
Sine 


.15043 
Cosin 


.98769 


.17365 


1.98481 




/ 


Cosin 


Sine 


Sine 


Cosin 


Sine 


84* 


83» 


82' 


81° 


80° 



202 



NATURAL SINES AND COSINES. 



/ 
~0 


10° 


11° 


12° 


13° 


14° 


/ 

60 


Sine 
717365 


! Cosin 
'798481 


Sine 
.19081 


Cosin 
.98163 


Sine 
.20791 


Cosin 

.97815 


Sine 
.22495 


Cosin 


Sine 


Cosin 


.97437 


.24192 


.97030 


1 


.17393 


.98476 


.19109 


.98157 


.20820 


.97809 


.22523 


.97430 


.24220 


.97023 


59 


2 


.17422 


.98471 


.19138 


.98152 


.20848 


.97803 


.22552 


.97424 


.24249 


.97015 


58 


3 


.17451 


.98166 


.19167 


.98146 


.20877 


.97797 


.22580 


.97417 


.24277 


.97008 


57 


4 


.17479 


.98461 


.19195 


.98140 


.20905 


.97791 


.22608 


.97411 


.24305 


.97001 


56 


5 


.17503 


.93455 


.19224 


.93135 


.20933 


.977&4 


.22637 


.97404 


.24333 


.96994 


55 


6 


.17537 


.98450 


.19252 


.98129 


.20962 


.97778 


.22665 


-97398 


.24362 


.96987 


54 


7 


.17565 


.93445 


.19231 


.98124 


.20990 


.97772 


.22693 


.97391 


.24390 


.96980 


53 


8 


.17591 


.98440 


.19309 


.93118 


.21019 


.97766 


.22722 


.97384 


.24418 


.96973 


52 


9 


.17623 


.93435 


.19333 


.93112 


.21047 


.97760 


.22750 


.97378 


.24446 


.96966 


51 


10 


.17651 


.98430 


.19366 


.98107 


.21076 


.97754 


.22778 


.97371 


.24474 


.96959 


50 


11 


.17680 


.98425 


.19395 


.98101 


.21104 


.97748 


.22807 


.97365 


.24503 


.96952 


49 


12 


.17703 


.98420 


.19423 


.93093 


.21132 


.97742 


.22835 


.97:358 


.24531 


.96945 


48 


13 


.17737 


.98414 


.19452 


.98C90; 


.21161 


.97735 


.22863 


.97351 


.24559 


.96937 


47 


14 


.17763 


.98109 


.19481 


.93084; 


.21189 


.97729 


.22892 


.97345 


.24587 


.96930 


46 


15 


.17794 


.98404 


.19509 


.98079 


.21218 


.97723 


.22920 


.97338 


.24615 


.96923 


45 


16 


.17823 


.98399 


.19533 


.98073 


.21246 


.977171 


.22948 


.97331 


.24644 


.96916 


44 


17 


.17852 


.98394 


.19566 


.98067 


.21275 


.97711 


.22977 


.97325 


.24672 


.96909 


43 


18 


.17880 


.98389 


.19595 


.98061 


.21303 


.97705 


.23005 


.97318 


.24700 


.96902 


42 


19 


.17909 


.98383 


.19623 


.98056 


.21331 


.97698 


.23033 


.97311 


.24728 


.96894 


41 


20 


.17937 


.98378 


.19652 


.93050 


.21360 


.97692 


.23062 


.97304 


.24756 


.96887 


40 


21 


.17960 


.93373 


.19680 


.93044 


.21388 


.97686 ! 


.23090 


.97298 


.24784 


.96880 


39 


22 


.17995 


.98363 


.19709 


.98039 


.21417 


.97680! 


.23118 


.97291 


.24813 


.96873 


S3 


23 


.18023 


.98362 


.19737 


.93033 


.21445 


.97673 


.23146 


.97284 


.24841 


.96866 


37 


24 


.18052 


.98:357 


.19763 


.93027 


.21474 


.976671 


.23175 


.97278 


.24869 


.96858 


36 


25 


.18081 


.98352 


.19794 


.93021! 


.21502 


.97661! 


.23203 


.97271 


.24897 


.96851 


35 


26 


.18109 


.93347 


.19323 


.98016: 


.21530 


. 97655 


.23231 


.97264 


.24925 


.96844 


34 


27 


.18133 


.98341 


.19851 


.93010 


.21559 


.97648 


.23200 


.97257 


.24954 


.96837 


33 


28 


.18166 


.98336 


.19880 


.93004 


.21587 


.97612 


.23288 


.97251 


.24982 


.96829 


32 


29 


.18195 


.93331 


.19908 


.97993 


.21616 


.97633 


.23316 


.97244 


.25010 


.96822 


31 


30 


.18224 


.98325 


.19937 


.97992 


.21644 


.97630 


.23345 


.97237 


.25038 


.96815 


30 


31 


.18252 


.98320 


.19965 


.97987 


.21672 


.976231 


.23373 


.97230 


.25060 


.96807 


29 


32 


.18281 


.98315 


.19994 


.97931 


.21701 


.97617 


.23401 


.97223 


.25094 


.96800 


28 


33 


.18309 


.98310 


.20022 


.97975 


.21729 


.97611! 


.23429 


.97217 


.25122 


.96793 


27 


34 


.18338 


.93304 


.20051 


.97989 


.21758 


.97301 


.23458 


.97210 


.25151 


.96786 


2(j 


35 


.18367 


.98299 


.20079 


.97933 


.21783 


.97593 


.23485 


.97203! 


.25179 


.96778 


25 


36 


.18395 


.93294 


.2J103 


.97953 


.21814 


.97592 


.23514 


.97196! 


.25207 


.96771 


24 


37 


.18424 


.98238 


.20136 


.97952 


.21843 


.97535 


.23542 


.97189; 


.25235 


.96764 


23 


38 


.18152 


.98283 


.20165 


.97946' 


.21871 


.97570 


.23571 


.97182' 


.25263 


.96756 


22 


39 


.18481 


.98277 


.20193 


.979401 


.21899 


.97573 


.23599 


.97176 


.25291 


.96749 


21 


40 


.18509 


.98272 


.20222 


.97934 


.21928 


.97566 


.23627 


. 97169 j 


.25320 


.96742 


20 


41 


.18538 


.98267 


.20250 


.97928 


.21956 


.97560' 


.23656 


.97162' 


.25348 


.96734 


19 


42 


.18567 


.98261 


.20279 


.97922J 


.219&5 


.97553 


.23684 


.97155 


.25376 


.96727 


18 


43 


.18595 


.98256 


.20307 


.97916 


.22013 


. 97547 • 


.23712 


.97148: 


.25404 


.96719 


17 


44 


.18624 


.98250 


.20336 


.97910 


.22041 


.97541 


.23740 


.97141! 


.25432 


.96712 


16 


45 


.18652 


.98245 


.20364 


.97905 


.22070 


.97534 


.23769 


.97134! 


.25460 


.96705 


15 


46 


.18681 


.98240 


.20393 


.97899! 


.22098 


.97528 


.23797 


.97127! 


.25488 


.96697 


14 


47 


.18710 


.98234 


.20421 


.97893| 


.22126 


.97521 


.23825 


.97120; 


.25516 


.96690 


13 


48 


.18738 


.93229 


.20450 


.97837 


.22155 


.97515! 


.23853 


.97113! 


.25545 


.96682 


12 


49 


.18767 


.98223 


.20478 


.97831 


.22183 


.97508! 


.23882 


.97106 


.25573 


.96675 


11 


50 


.18795 


.93218 


.20507 


.97875 


.22212 ; 


. 97502 j 


.23910 


.97100' 


.25601 


.96667 


10 


51 


.18824 


.98212 


.20535 


.97869 


.22240 


.97496' 


.23938 


.97093! 


.25629 


.96660 


9 


52 


.18852 


.98207 


.20563 .97863! 


.22268 


.97489 


.23966 


. 97086 i 


.25657 .96653 


8 


53 


.18881 


.98201 


.20592 '.978571 


.22297 


.97483! 


.23995 


.97079) 


.25685 .96645 


7 


54 


.18913 


.98195 


.20320!. 97851 j 


.22325 


.97476 


.24023 


.97072 


.25713 .96638 


6 


55 


.18938 


.98190 


.20649S.97845! 


.22:353 


.97470 


.24051!. 97065' 


.25741 .96630 


5 


56 


. 18967 * 


.981851 


. 20677 L 97839| 


.22382 


.97463 


. 24079 i. 97058 


.25769 .96623 


4 


57 


. 18995 ' 


.98179 


.20706 '.97833! 


.22410 


.97457 


.24108 


.97051 


.25798 .96615 


3 


53 


.19024 


.98174 


.20734 .97827! 


.22438 


.97450 


.24136 


.97044! 


.25826 .96608 


2 


59 


.19052 


.981681 


.20763 .97821 


.22467 


.97444 


.24164 


.97037' 


.25854 .96600 


1 




.19081; 


.98163 


.20791 .97815 


.22495 


.97437 


.24192 
Cosin 


.97030 ; 
Sine 


.25882 .96593 




/ 


Cosin 


Sine 


Cosin 1 Sine 


Cosin 


Sine 


Cosin Sine 


79° 


78° 


77° 


76° 


75° 



N A TUBAL SIXES AND COSINES. 



V 




15° 


16° 


17° 


18° 


19° 


1 
60 


Sine 


Cosin 


Sine 
.27564 


Cosin 


Sine 


Cosin 
.95630 


j Sine 
.30902 


Cosin 


Sine 
.32557 


Cosin 


.25882 


.96593 


.96126 


.29237 


.95106 


.94552 


1 


.25910 


.96585 


.27592 .96118 


.29265 


.95622; 


.30929 


.95097 


.32584 


.94542 


59 


2 


.25938 


.96578 


.27620 .96110 


.29293 


.956131 


.30957 


.95088 


.32612 


.94533 


58 


3 


.25966 


.96570! 


.27648 .96102 


.29321 


.95605: 


.30985 


.95079 


.32639 


.94523 


57 


4 


.25994 .965621 


.27676 .96094 


.29348 


.95596' 


.31012 


.95070 


.32667 


.94514 


56 


5 


.26022 .96555 i 


.27704 .96086| 


.29376 


.95588' 


.31040 


.95061 


.32694 


.94504 


55 


6 


.260501.965471 


.27731 .96078! 


.29404 


.95579 


.31068 


.95052 


.32722 


.94495 


54 


7 


.26079!. 96540 ! 


.27759*96070 


.29432 


.95571 


; .31095 


.95043 


.32749 


.94485 


53 


8 


.261071.96532! 


.27787 .96062! 


.29460 


.95562 


.31123 


.95033 


.32777 


.94476 


52 


9 


.261351.96524! 


.27815 .96054! 


.29487 


.95554 


.31151 


.95024 


.32804 


.94466 


51 


10 


.26163 .965171 


.27843 ,.96046 


.29515 


.95545 


.31178 


.95015 


.32832 


.94457 


50 


11 


.261911.96509' 


.27871 !. 96037 


.29543 


.95536 


.31206 


.95006 


.32859 


.94447 


49 


12 


.26219 .96502: 


.27899 .96029 ! 


.29571 


.95528 


.31233 


.94997 


.32887 


.94438 


48 


13 


.26247 .96494; 


.27927. 96021 : 


.29599 


.95519! 


.31261 


.94988 


.32914 


.94428 


47 


14 


.26275;. 96486| 


.27955 .96013: 


.29626 


.95511 


' .31289 


.94979 


.32942 


.94418 


46 


15 


.26303 .96479 


.27983 .96005i 


.29654 


.95502 


.31316 


.94970 


.32969 


.94409 


45 


16 


.263311. 96471 i 


.28011!. 95997 


.29682 


.95493 


.31344 


.94961 


.32997 


.94399 


44 


17 


.26359 .96463 


.28039 .95989 


.29710 


.95485! 


.31372 


.94952 


.33024 


.94390 


43 


18 


.26387. 96456 


.28067 


.95981 


.29737 


.95476! 


.31399 


.94943 


.33051 


.94380 


42 


19 


.26415 .96448 


.28095 


.95972 


.29765 


.95467 


! .31427 


.94933 


1 .33079 


.94370 


41 


20 


.26443 


.96440, 


.28123 


.95964 


.29793 


.95459 


.31454 


.94924 


.33106 


.94361 


40 


21 


.26471 


. 96433 ! 


.28150 .95956 


.29821 


.95450 


.31482 


.94915 


.33134 


.94351 


39 


22 


.26500 


.96425 


.28178;. 95948 


.29849 


.95441 


! .31510 


.94906 


.33161 


.94342 


38 


23 


.26528 


.96417, 


.28206!. 95940 


.29876 


.95433 


1 .31537 


.94897 


j .33189 


.94332 


37 


24 


.26556 


.96410; 


.282^4 .95931 


: .29904 


.95424 


! .31565 


.94888 


! .33216 


.94322 


36 


25 


.26584 


.96402 


.28262 .95923 


.29932 


.95415 


; .31593 


.94878 


.33244 


.94313 


35 


26 


.26612 


.96394 


.28290 .95915 


! .29960 


.95407 


i .31620 


.94869 


; .33271 


.94303 


34 


27 


.26640 


.96386 


.28318 .95907 


.29987 


.95398 


i .31648 


.94860 


; .33298 


.94293 


33 


28 


.26668 


.96379 


.28346 .95898 


.30015 


.95389 


! .31675 


.94851 


1.33326 


.942S4 


32 


29 


.26696 


.96371 


.28374 .95890 


.30043 


.95:380 


.31703 


.94842 


! .33353 


.94274 


31 


30 


.26724 


.963631 


.28402;. 95882 


.30071 


.95372 


.31730 


.94832 


j .33381 


.9426-4 


30 


31 


.26752 


. 96355 \ 


.28429 .95874 


.30098 


.95363 


.31758 


.94823 


'.33408 


.94254 


29 


32 


.26780 


.96347 


.28457|.95865 


.30126 


.95354 


.31786 


.94814 


.33436 


.94245 


28 


33 


.26808 


.96340 


.28485 .95857 


.30154 


.95345 


! .31813 


.94805 


; .33463 


.94235 


27 


34 


.26836 


.96332 


.28513;. 95849 


.30182 


.95337 


! .31841 


.94795 


1.33490 


.94225 


26 


35 


.26864 


.96324 


.28541;.95841 


.30209 


.95328 


1 .31868 


.94780 


.33518 


.94215 


25 


36 


.26892 


.96316 


.28569 .95832 


! .30237 


.95319 


i .31896 


.94777 


.33545 


.94203 


24 


37 


.26920 


.96308 


.285971.95824 


1 .30265 


.95310 


! .31923 


.94768 


.33573 


.94190 


23 


38 


.269481.96301 


.28625;. 95816 


! .30292 


.95301 


1 .31951 


.94758 


.33600 


.94186 


22 


39 


.26976 


.96293 


.286521.95807 


i .30320 


.95293 


! .31979 


.94749 


j .33627 


.94176 


21 


40 


.27004 


.96285: 


.28680,. 95799 


.30348 


.95284 


".32006 


.94740 


J.33655 


.94167 


20 


41 


.27032 


.96277 


.28708 .95791 


.30376 


.95275 


1 .32034 


.94730 


! .33682 


.94157 


19 


42 


.27060 


.96269 


.28736 .95782 


• .30403 


.95266 


1 .32061 


.94721 


.33710 


.94147 


18 


43 


.27088 


.96261 


.28764 .95774 


1.30431 


.95257 


[.32089 


.94712 


: .33737 


.94137 


17 


44 


.27116 


.96253 


.28792 .95766 


1.30459 


.95248 


! .32116 


.94702 


.33764 


.94127 


16 


45 


.27144 


.96246 


.28820 .95757 


| .30486 


.95240, 


.32144 


.94693 


1 .33792 


.94118 


15 


46 


.27172 


.96238 


.288471.95749 


! .30514 


.952311 


.32171 


.94684 


i .33819 


.94108 


14 


47 


.27200 


.96230 


.288751.95740 


1 .30542 


.952221 


.32199 


.94674 


[.33846 


.94098 


13 


48 


.27228 


.96222 


.28903 .95732 


! .30570 


.95213! 


.32227 


.94665 


1.33874 


.94088 


12 


49 


.27256 


.96214 


.28931 .95724 


i .30597 


.95204, 


.32254 


.94656 


! 33901 


.94078 


11 


50 


.27284 


.96206, 


,28959 ..95715 


.30625 


.95195 


.32282 


.9464d 


1.33929 


.94068 


10 


51 


.27312 


.96198' 


.28987 .95707 


.30653 


.95186! 


.32309 


.94637 


! .33956 


.94058 


9 


52 


.27340 


.96190 


.290151.95698 


.30680 


.95177; 


.32337 


.94627 


j .33983 


.94049 


8 


53 


.27368 


.96182 


.29042 .95690 


.30708 


.95168; 


.32364 


.94618 


■ .34011 


.94039 


7 


54 


.27396 


.96174 


.290701.95681 


.30736 


.95159 


.32392 


.94609 


! .34038 


.94029 


6 


55 


.27424 


.96166 


.29098 .95673 


.30763 


.95150- 


.32419 


.94599 


I .34065 


.94019 


5 


56 


.27452 


.96158 


.29126J.95664 


.30791 


.95142 


.32447 


.94590 


1 .34093 


.94009 


4 


57 


.27480 


.96150 


.291 54L 95656 


.30819 


.95133 


.32474 


.93580 


: .34120 


.93999 


3 


58 


.27508 .96142 


.29182 .95647 


.30846 


.95124 


.32502 


.94571 


1 .34147 


.93989 


2 


59 


.27536 


.96134 


.29209 .95639 


.30874 


.95115 


.32529 


.94561 


.34175 


.93979 


1 


60 

/ 


.27564 


.96126 
Sine 


.292371.95630 


.30902 


.95106 


.32557 
Cosin 


.94552 
Sine 


.34202 


.93969 



/ 


Cosin 


Cosinl Sine 


Cosin 


Sine i 


Cosin 


Sine 


74° 


73° 


72° 


71° 


70° 





204 



NATURAL SINES AND COSINES. 



t 

"o 


20° 


21° 


22° 


23° 


24° 


/ 
60 


Sine 


Cosin 


Sine 
.35837 


Cosin 


Sine 


Cosin 


Sine 


Cosin 
792050 


Sine Cosin 


.34202 


.93969 


.93358 


.37461 


.92718 


.39073 


T40674 


.91355 


1 


.34229 


.93959 


.35864 


.93348 


.37488 


.92707 


.39100 


.92039 


.40700 


.91343 


59 


2 


.34257 


.93949 


.35891 


.93337 


.37515 


.92697 


.39127 


.92028 


.40727 


.91331 


58 


3 


.34284 


.93939 


.35918 


.93327 


.37542 


.92686 


.39153 


.92016 


.40753 


.91319 


5? 


4 


.34311 


.93929 


.35945 


.93316 


.37569 


.92675 


.39180 


.92005 


.40780 


.91307 


56 


5 


.34339 


.93919 


.35973 


.93306 


.37595 


.92664 


.39207 


.91994 


.40806 


.91295 


55 


6 


.34366 


.93909 


.36000 


.93295 


.37622 


.92653 


.39234 


.91982 


.40833 


.91283 


54 


7 


.34393 


.93899 


.36027 


.93285 


.37649 


.92642 


.39260 


.91971 


.40860 


.91272 


53 


8 


.34421 


.93889 


.36054 


.93274 


.37676 


.92631 


.39287 


.91959 


.40886 


.91260 


52 


9 


.34448 


.93879 


.36081 


.93264 


.37703 


.92620 


.39314 


.91948 


.40913 


.91248 


51 


10 


.34475 


.93869 


.36108 


.93253 


.37730 


.92609 


.39341 


.91936 


.40939 


.91236 


50 


11 


.34503 


.93859 


.36135 


.93243 


.37757 


.92598 


.39367 


.91925 


.40966 


.91224 


49 


12 


.34530 


.93849 


.36162 


.93232 


.37784 


.92587 


.39394 


.91914 


.40992 


.91212 


48 


13 


.34557 


.93839 


.36190 


.93222 


.37811 


.92576 


.39421 


.91902 


.41019 


.91200 


47 


14 


.34584 


.93829 


.36217 


.93211 


.37838 


.92565 


.39448 


.91891 


.41045 


.91188 


46 


15 


.34612 


.93819 


.36244 


.93201 


.37865 


.92554 


.39474 


.91879 


.41072 


.91176 


45 


16 


.34639 


.93809 


.36271 


.93190 


.37892 


.92543 


.39501 


.91868 


.41098 


.91164 


44 


17 


.34666 


.93799 


.36298 


.93180 


.37919 


.92532 


.39528 


.91856 


.41125 


.91152 


43 


18 


.34694 


.93789 


.36325 


.93169 


.37946 


.92521 


.39555 


.91845 


.41151 


.91140 


42 


19 


.34721 


.93779 


.36352 


.93159 


.37973 


.92510 


.39581 


.91833 


.41178 


.91128 


41 


20 


.34748 


.93769 


.36379 


.93148 


.37999 


.92499 


.39608 


.91822 


.41204 


.91116 


40 


21 


.34775 


.93759 


.36406 


.93137 


.38026 


.92488 


.39635 


.91810 


.41231 


.91104 


39 


22 


.34803 


.93748 


.36434 


.93127 


.38053 


.92477 


.39661 


.91799 


.41257 


.91092 


38 


23 


.34830 


.93738 


.36461 


.93110 


.38080 


.92466 


.39688 


.91787 


.41284 


.91080 


37 


24 


.34857 


.93728 


.36488 


.93106 


.38107 


.92455 


.39715 


.91775 


.41310 


.91068 


36 


25 


.34884 


.93718 


.36515 


.93095 


.38134 


.92444 


.39741 


.91764 


.41337 


.91056 


35 


26 


.34912 


.93708 


.36542 


.93084 


.38161 


.92432 


.39768 


.91752 


.41363 


.91044 


34 


27 


.34939 


.93698 


.36569 


.93074 


.38188 


.92421 


.39795 


.91741 


.41390 


.91032 


33 


28 


.34966 


.93688 


.36596 


.93063 


.38215 


.92410 


.39822 


.91729 


.41416 


.91020 


32 


29 


.34993 


.93077 


.36623 


.93052 


.f.8241 


.92399 


.3984S 


.91718 


.41443 


.91008 


31 


30 


.35021 


.93667 


.36650 


.93042 


.38268 


.92388 


.39875 


.11706 


.41469 


.90996 


30 


31 


.35048 


.9365? 


.36677 


.93031 


.38295 


.92377 


.39902 


.91G94 


.41496 


.90984 


29 


32 


.35075 


.93647 


.36704 


.93020 


.38322 


.92366 


.39928 


.916£3 


.41522 


.90972 


23 


33 


.35102 


.93637 


.36731 


.93010 


.38349 


.92355 


.39955 


.91671 


.41549 


.90960 


27 


34 


.35130 


.93626 


.36758 


.92999 


.38376 


.92343 


.39982 


.91660 


.41575 


.90948 


26 


35 


.35157 


.93616 


.36785 


.92988 


.38403 


.92332 


.40008 


.91648 


.41602 


.90936 


25 


36 


.35184 


.93606 


.36812 


.92978 


.38430 


.92321 


.40035 


.91636 


.41628 


.90924 


24 


37 


.35211 


93596 


.36839 


.92967 


.38456 


.92310 


.40062 


.91C25 


.41655 


.90911 


23 


38 


.35239 


.93585 


.36867 


.92956 


.38483 


.92299 


.40088 


.91613 


.41681 


.90899 


22 


39 


.35266 


.93575 


.36894 


.92945 


.38510 


.92287 


.40115 


.91601 


.41707 


.90887 


21 


40 


.35293 


.93565 


.36921 


.92935 


.38537 


.92276 


.40141 


.91590 


.41734 


.90875 


20 


41 


.35320 


.93555 


.36948 


.92924 


.38564 


.92265 


.40168 


.91578 


.41760 


.90863 


19 


42 


.35347 


.93544 


.36975 


.92913 


.38591 


.92254 


.40195 


.91566 


.41787 


.90851 


18 


43 


.35375 


.93534 


.37002 


.92902 


.38617 


.92243 


.40221 


.91555 


.41813 


.90839 


17 


44 


.35402 


.93524 


.37029 


.92892 


.38644 


.92231 


.40248 


.91543 


.41840 


.90826 


16 


45 


.35429 


.93514 


.37056 


.92881 


.38671 


.92220 


.40275 


.91531 


.41866 


.90814 


15 


46 


.35456 


.93503 


.37083 


.92870 


.38698 


.92209 


.40301 


.91519 


.41892 


.90802 


14 


47 


.35484 


.93493; 


.37110 


.92859 


.38725 


.92198 


.40328 


.91508 


.41919 


.90790 


13 


48 


.35511 


.93483 


.37137 


.92849 


.38752 


.92186 


.40355 


.91496 


.41945 


.90778 


12 


49 


.35538 


.93472 


.37164 


.92838 


.38778 


.92175 


.40381 


.91484 


.41972 


.90766 


11 


50 


.35565 


.93462; 


.37191 


.92827 


.38805 


.92164 


.40408 


.91472 


.41998 


.90753 


10 


51 


.35592 


.93452! 


.37218 


.92816 


.38832 


.92152 


.40434 


.91461 


.42024 


.90741 


9 


52 


.35619 


.93441; 


.37245 


.92805 


.38859 


.92141 


.40461 


.91449 


.42051 


.90729 


8 


53 


.35647 


.93431 


.37272 


.92794 


.38886 


.92130 


.40488 


.91437 


.42077 


.90717 


7 


54 


.35674 


.93420 


.37299 


.92784 


.38912 


.92119 


.40514 


.91425 


.42104 


.90704 


6 


55 


.35701 


.93410; 


.37326 


.92773 


.38939 


.92107 


.40541 


.91414 


.42130 


.90692 


5 


56 


.35728 


.93400' 


.37353 


.92762 


.38966 


.92096 


.40567 


.91402 


.42156 


.90680 


4 


57 


.35755 


.93389! 


.37380 


.92751 


.38993 


.92085 


.40594 


.91390 


.42183 


.90668 


3 


58 


.35782 


.93379' 


.37407 


.92740 


.39020 


.92073 


.40621 


.91378 


.42209 


.90655 


2 


59 


.35810 


.93368! 


.37434 


.92729 


.39046 


.92062 


.40647 


.91366 


.42235 


.90643 


1 


60 

/ 


.35837 


.93358! 


.37461 
Cosin 


.92718 


.39073 
Cosin 


.92060 


.40674 


.91355 


.42262 


.90631 


J) 

/ 


Cosin 


Sine 


Sine 


Sine 


Cosin 


Sine 


Cosin 


Sine 


69° 


68° 


67° 


66° 


65° 



205 



NATURAL SINES AND COSINES. 



~o 


25° 


26° 


27° 


28° 


29° 


/ 

60 


Sine 


Cosin 


Sine 


Cosin 
.89879 


Sine 


Cosin 
.89101 


Sine 


Cosin 


Sine 


Cosin 

.87462 


.42262 


.90631 


.43837 


.45399 


.46947 


.88295 


.48481 


1 


.42288 


.90818 


.43863 


.89867 


.45425 


.89087 


.46973 


.88281 


.48506 


.87448 


59 


2 


.42315 


.90606 


.43889 


.89854 


.45451 


.89074 


.46999 


.88267 


.48532 


.87434 


58 


3 


.42341 


.90594 


.43916 


.89841 


.45477 


.89061 


.47024 


.88254 


.48557 


.87420 


57 


4 


.42367 


.90582 


.43942 


.89828 


.45503 


.89048 


.47050 


.8S240 


.48583 


.87406 


56 


5 


.42394 


.90569 


.43968 


.89816 


.45529 


.89035 


.47076 


.88226 


.48608 


.87391 


55 


6 


.42420 


.90557 


.43994 


.89803 


.45554 


.89021 


.47101 


.88213 


.48634 


.87377 


54 


7 


.42446 


.90545 


.44020 


.89790 


.45580 


.89008 


.47127 


.88199 


.48659 


.87363 


53 


8 


.42473 


.90532 


.44046 


.89777 


.45606 


.8C995 


.47153 


.88185 


.48684 


.87349 


52 


9 


.42499 


.90520 


.44072 


.89764 


.45632 


.88981 


.47178 


.88172 


.48710 


.87335 


51 


10 


.42525 


.90507 


.44098 


.89752 


.45658 


.88968 


.47204 


.88158 


.48735 


.87321 


50 


11 


.42552 


.90495 


.44124 


.89739 


.45684 


.88955 


.47229 


.88144 


.48761 


.87306 


49 


12 


.42578 


.90483 


.44151 


.89726 


.45710 


.88942 


.47255 


.88130 


.48786 


.87292 


48 


13 


.42604 


.90470 


.44177 


.89713 


.45736 


.88928 


.47281 


.88117 


.48811 


.87278 


47 


14 


.42631 


.90458 


.44203 


.89700 


.45762 


.88915 


.47306 


.88103 


.48837 


.87264 


46 


15 


.42657 


.90446 


.44229 


.89687 


.45787 


.88902 


.47332 


.88089 


.48862 


.87250 


45 


16 


.42683 


.90433 


.44255 


.89674 


.45813 


.88888 


.47358 


.88075 


.48888 


.87235 


44 


17 


.42709 


.90421 


.44281 


.89662 


.45839 


.88875 


.47383 


.88062 


.48913 


.87221 


43 


18 


.42736 


.9040S 


.44307 


.89649] 


.45865 


.88862 


.47409 


.88048 


.48938 


.87207 


42 


19 


.42762 


.90396 


.44333 


.89636! 


.45891 


.88848 


.47434 


.88034 


.43964 


.87193 


41 


20 


.42788 


.90383 


.44359 


.896231 


.45917 


.88835 


.47460 


.88020 


.48989 


.87178 


40 


21 


.42815 


.90371 


.44385 


.89610 


.45942 


.88822 


.47486 


.88006 


.49014 


.87164 


39 


22 


.42841 


.90353 


.44411 


.89597 


.45963 


.88803 


.47511 


.87993 


.49040 


.87150 


38 


23 


.42867 


.90346 


.44437 


.89584! 


.45994 


.88795 


.47537 


.87979 


.49065 


.87136 


37 


24 


.42894 


.90334 


.44464 


.89571 i 


.46020 


.83782 


.47502 


.87965 


.49090 


.87121 


36 


25 


.42920 


.90321 


.44490 


.89558: 


.46046 


.88768 


.47588 


.87951 


.49116 


.87107 


35 


26 


.42946 


.90309 


.44516 


.89545 


.46072 


.88755 


.47614 


.87937 


.49141 


.87093 


34 


27 


.42972 


.90296 


.44542 


.89532 


.46097 


.88741 


.47G39 


.87923 


.49166 


.87079 


33 


25 


.42999 


.90284 


.44568 


.89519 


.46123 


.88728 


.47665 


.87909 


.49192 


.87064 


32 


2.") 


.43025 


.9-271 


.44594 


.89506 


.46149 


.83715 


.47G90 


.87896 


.49217 


.87050 


31 


30 


.43051 


.90259 


.44620 


.89493 


.46175 


.88701 


.47716 


.87882 


.49242 


.87036 


30 


31 


.43077 


.90246 


.44646 


.89480 


.46201 


.88688 


.47741 


.87868 


.49268 


.87021 


29 


32 


.43104 


.90233 


.44672 


.83467 


.46226 


.83074 


.477G7 


.87854 


.49293 


.87007 


23 


33 


.43130 


.90221 


.44698 


.89454 


.46252 


.88661 


.47793 


.87840 


.49318 


.86993 


27 


34 


.43156 


.90203 


.44724 


.89441 


.46278 


.88647 


.47818 


.87826 


.49344 


.86978 


23 


35 


.43182 


.90196 


.44750 


.89428 


.46304 


.88634 


.47844 


.87812 


.49369 


.86964 


25 


36 


.43209 


.90183 


.44776 


.89415 


.46330 


.88620 


.47869 


.87798 


.49394 


.86949 


24 


37 


.43235 


.90171 


.44802 


.89402 


.46355 


.88607 


.47895 


.87784 


.49419 


.86935 


23 


33 


.43261 


.90153 


.44828 


.89389 


.46381 


.88593 


.47920 


.87770 


.49445 


.86921 


22 


39 


.43287 


.90146 


.44854 


.89376 


.46407 


.88580 


.47940 


.87756 


.49470 


.86906 


21 


40 


.43313 


.90133 j 


.44880 


.89363 


.46433 


.88566 


.47971 


.87743 


.49495 


.86892 


20 


41 


.43340 


.90120] 


.44906 


.89350 


.46458 


.88553 


.47997 


.87729 


.49521 


.86878 


19 


42 


.43366 


.90103 ! 


.44932 


.89337 


.46484 


.83539 


.48022 


.87715 


.49546 


.86863 


18 


43 


.43392 


.90095 


.44958 


.89324 


.46510 


.88526 


.48048 


.87701 


.49571 


.86849 


17 


44 


.43418 


.900S2 


.44984 


.89311 


.46536 


.88512 


.48073 


.87687 


.49596 


.86834 


16 


45 


.43445 


.90070; 


.45010 


.89298 


.46561 


.88499 


.48099 


.87673 


.49622 


.86820 


15 


46 


.43471 


.90057 i 


.45036 


.89285 


.465S7 


.88485 


.48124 


.87659 


.49647 


.86805 


14 


47 


.43497 


.900451 


.45052 


.89272 


.46613 


.88472 


.48150 


.87645 


.49672 


.86791 


13 


48 


.43523 


.90032 


.45088 


.89259 


.46639 


.88458 


.48175 


.87631 


.49697 


.86777 


12 


49 


.43549 


.90019 


.45114 


.89245 


.46664 


.88445 


.48201 


.87617 


.49723 


.86762 


11 


50 


.43575 


.90007 


.45140 


.89232 


.46690 


.88431 


.48226 


.87603 


.49748 


.86748 


10 


51 


.43602 


.89994 1 


.45166 


.89219 


.46716 


.88417 


.48252 


.87589 


.49773 


.86733 


9 


52 


.43628 


.89981 


.45192 


.89206 


.46742 


.88404 


.48277 


.87575 


.49798 


.86719 


8 


53 


.43654 


.89038; 


.45218 


.89193 


.46767 


.88390 


.48303 


.87561 


.49824 


.86704 


7 


54 


.43680 


.89956 


.45243 


.89180 


.46793 


.88377 


.48328 


.87546 


.49849 


.86690 


6 


55 


.43706 


.89943 


.45269 


.89167 


.46819 


.88363 


.48354 


.87532 


.49874 


.86675 


5 


56 


.43733 


.89930 


.45295 


.89153 


.46844 


.88349 


.48379 


.87518 


.49899 


.86661 


4 


57 


.43759 


.89918 


.45321 


.89140 


.46870 


.88336 


.48405 


.87504 


.49924 


.86646 


3 


58 


.43785 


.89905 


.45347 


.89127 


.46896 


.88322 


.48430 


.87490 


.49950 


.86632 


2 


59 


.43811 


.89892 


.45373 


.89114 


.46921 


.88308 


.48456 


.87476 


.49975 


.86617 


1 


60 

/ 


.43837 


.89879 


.45399 


.89101 


.46947 


.88295 


.48481 


.87462 


.50000 


.86603 




/ 


Cosin 


Sine 


Cosin 


Sine 


Cosin 


Sine 


Cosin 


Sine 


Cosin | 


Sine 


64° 


63° 


62° 


61° 


60° 



206 



NATURAL SINES AND COSINES. 



"o 


30° 


31 


o 


32" 


33 


• 


34 


o 


/ 
60 


Sine 
.50000 


Cosin 

.86603 


Sine 


Cosin 


Sine ! Cosin 

.529921.84805 


Sine 1 Cosin 
.54464 .83867 


Sine 1 Cosin 


.51504 


.85717 


.55919 .82904 


1 


.50025 


.86588 


.51529 


.85702 


.53017 .84789 


.54488 


.83851 


.55943 .82887, 


59 


2 


.50050 


.86573 


.51554 


.85687 


.53041 .84774 


.54513 


.83835 


.55968 .82871; 


58 


3 


.50076 


.86559 


.51579 


.85672 


.53066!. 84759 


.54537 


.83819 


.559921.82855^ 


57 


4 


.50101 


.86544 


.51604 


.85657 


.530911.84743 


.54561 ; 


.83804 


.56016! 


. 82839 : 


56 


5 


.50126 


.86530 


.51628 


.85642 


.53115 .84728 


.54586 


.83788 


.56040; 


.82822* 


55 


6 


.50151 


.86515 


.51653 


.85627 


.531401.84712 


.54610 


.83772: 


.56064: 


. 82806 ; 


54 


7 


.50176 


.86501 


.51678 


.85612 


.53164i.84697 


.54635 


.83756! 


.56088! 


.82790 


53 


8 


.50201 


.86486 


.51703 


.85597 


.531891.84681 


.54659' 


.83740: 


.56112 


.82773 


52 


9 


.50227 


.86471 


.51728 


.85582 


.53214 J.84666 


.54683 


.83724; 


.561361 


.82757 


51 


10 


.50252 


.86457 


.51753 


.85567 


. 53238 j. 84650 


. 54708 ( 


.837o8| 


.56160 


.82741 


50 


11 


.50277 


.86442 


.51778 


.85551 


.532631.84635* 


.54732 


.83692' 


.56184 


.82724 


49 


12 


.50302 


.86427 


.51803 


.85536 


.532881.84619; 


.54756 


.83676 


.56208 


.82708 


48 


13 


.50327 


.86413 


.51823 


.85521 


.53312!. 84604 


.54781 ! 


.836601 


.56232 


.82692 


47 


14 


.50352 


.86398 


.51852 


.85506 


.53337 


.£4588 


.54805' 


.83645 


.56256 


.82675 


46 


15 


.50377 


.86384 


.51877 


.85491 


.53361 


.84573 


.54829 


.83629; 


.56280 


.82659 


45 


16 


.50403 


.86369 


.51902 


.85476 


.53386 


.84557 


.54854 


.83613! 


.56305 


.82643 


44 


17 


.50428 


.86354 


.51927 


.85461 


.53411 


.84542 


.54878 


.835971 


.56329 


.82626 


43 


18 


.50453 


.86340 


.51952 


.85446 


.53435 


.84526 


.54902 


.83581! 


.56353 


.82610 


42 


19 


.50478 


.86325 


.51977 


.85431 


.53460 


.84511 


.54927 


.83565! 


.56377 


.82593 


41 


20 


.50503 


.86310 


.52002 


.85416 


.53484 


.84495 


.54951 


.83549' 


.56401 


.82577 


40 


21 


.50528 


.86295 


.52026 


.85401 


.53509 


.84480' 


.54975 


.83533 


.56425 


.82561 


39 


22 


.50553 


.86281 


.52051 


.85385 


.53534 


.84464 


.54999 


.83517 


.56449 


.82544 


38 


23 


.50578 


.86286 


.52076 


.85370 


.53558 


.84448 


.55024 


.83501 


.56473 


.82528 


37 


24 


.50603 


.86251 


.52101 


.85355 


.53583 


.84433 


.55048 


.83485 


.56497 


.82511 


36 


25 


.50628 


.86237 


.52126 


.85340 


.53607 


.84417 


.55072 


.8*469 


.56521 


.82495 


35 


26 


.506.54 


.86222 


.52151 


.85325 


.53632 .84402 


.55097 


.83453 


.56545 


.82478 


34 


27 


.50679 


.86207 


.52175 


.85310 


.536561.84386 


.55121 


.83437 


.56569 


.82462 


33 


28 


.50704 


.86192 


.52200 


.85294 


.53681 .84370 


.55145 


.83421 


.56593 


.82446 


32 


29 


.50729 


.86178 


.52225 


.85279 


.53705 .84355 


.55169 


.83405 


.56617 


.82429 


31 


30 


.50754 


.86163 


.52250 


.85264 


.537301.84339 


.55194 


.83389 


.56641 


.82413 


30 


31 


.50779 


.86148 


.52275 


.85249 


. 53754 L&4324 


.55218 


.83373' 


.56665 


.82396 


29 


32 


.50804 


.86133 


.52299 


.852*4 


.53779;. 84303 


.55242 


.83356 


.56689 


.82380 


28 


33 


.50829 


.86119 


.52324 


.85218: 


.538041.84292 


.£5266 


.83340 


j .56713 


.82363 


27 


34 


.50854 


.86104 


.52349 


.85203; 


.53828 .84277 


.55291 


.83324 


I .56736 


.82347 


26 


35 


.50879 


.86089 


.52374 


.85188i 


.53853 .84261 


.55315 


.83308, 


.56760 


.82330 


25 


36 


.50904 


.86074 


.52399 


.85173 ' 


. 53877 1. 84245 


.55339 


.83292 


.56784 


.82314 


24 


37 


.50929 


.86059| 


.52423 


.85157 


.53902 .84230 


.55363 


.83276 


.56808 


.82297 


23 


38 


.50954 


.86045 


.52448 


.85142 


.£3926 
]o3951 


.84214 


.55388 


.83260 


.56832 


.82281 


22 


39 


.50979 


.86030 


.52473 


.85127 


.84198 


.55412 


.83244 


.56856 


.82264 


21 


40 


.51004 


.86015; 


.52498 


.85112 


.53975 


.84182 


.55436 


.83228, 


.56880 


.82248 


20 


41 


.51029 


. 86000 ' 


.52522 


.85096 


.54000 


.84167 


.55460 


.83212' 


! .56904 


.82231 


19 


42 


.51054 


.85985 


.52547 


.85081 


.54024 


.84151 


.55484 


.83195 


j .56928 


.82214 


18 


43 


.51079 


.85970 


.52572 


.85066 


.54049 


.84135 


.55509 


.83179 


! .56952 


.82198 


17 


44 


.51104 


.85956 


.52597 


.85051 


.54073 


.84120 


.55533 


.83163 


.56976 


.82181 


16 


45 


.51129 


.85941 


.52621 


.85035 


.54097 


.84104 


.55557 


.83147 


.57000 


.82165 


15 


46 


.51154 


.85926 


.52646 


.85020 


.54122 


.84088 


.55581 


.83131 


.57024 


.82148 


14 


47 


.51179 


.85911 


.53671 


.85005 


.54146 


.84072 


.55605 


.83115 


.57047 


.82132 


13 


48 


.51204 


.85896 


.52696 


.84989 


.54171 


.84057 


.55630 


.83098 


.57071 


.82115 


12 


49 


.51229 


.85881 


.52720 


.84974 


.54195 


.84041 


.55654 


.83082 


1 .57095 


.82098 


11 


50 


.51254 


.85866 


.52745 


.84959 


.54220 


.84025 


.55678 


.83066 


.57119 


.82082 


10 


51 


.51279 


.85851 


.52770 


.84943 


.54244 


.84009 


.55702 


.83050 


.57143 


.82065 


9 


52 


.51304 


.85836 


.52794 


.84928 


.54269 


.83994 


.55726 


.83034 


i .57167 


.82048 


8 


53 


.51329 


.85821 


.52819 


.84913 


.54293 


.83978 


.55750 


.83017 


! .57191 


.82032 


7 


54 


.51354 


.85806 


.52844 


.84897 


.54317 


.83962 


.55775 


.83001 


.57215 


.82015 


6 


55 


.51379 


.85792 


.52869 


.84882 


.54342 


.83946 


.55799 


.82985 


.57238 


.81999 


5 


56 


.51404 


.85777 


.52893 


.84866 


.54366 


.83930 


.55823 


.82969 


1 .57262 


.81982 


4 


57 


.51429 


.85762 


.52918 


.84851 


.54391 


.83915 


.55847 


.82953 


! .57286 


.81965 


3 


58 


.51454 


.85747 


.52943 


.84836 


.54415 


.83899 


.55871 


.82936 


.57310 


.81949 


2 


59 


.51479 


.85732 


.52967 


.84820 


.54440 


.83883 


! .55895 


.82920 


.57*34 


.81932 


1 


60 


.51504 


i. 85717 


.52992 


.84805 


.54464 
! Cosin 


.83867 
| Sine 


! .55919 


.82904 


.57358 
Cosin 


.81915 
Sine 



/ 


Cosin 


Sine 


Cosin 


Sine 


Cosin 


Sine 


59° 


58° 


57° 


56° 


55° 



207 



NATURAL SINES AND COSINES. 



1 

"o 


35° 


36° 


37° 


38° 


39° 


/ 
60 


Sine 


Cosin 


Sine 

758779 


Cosin 

.80902 


Sine 

.60182 


Cosin 

.79864 


Sine 
.61566 


Cosin 
.78801 


Sine 
.62932 


Cosin 


.57358 


.81915 


.77715 


1 


.57381 


.81899; 


.58802 


.80885' 


.60205 


.79846 


.61589 


.78783 


.62955 


.77696 


59 


2 


.57405 


.81882! 


.58826 


.80867! 


.60228 


.79829! 


.61612 


.78765 


.62977 


.77678 


58 


3 


.57429 


.81865 ' 


.58849 


.80850; 


.60251 


.79811! 


.61635 


.78747 


.63000 


.77660 


57 


4 


.57453 


.81848 


.58873 


.80833! 


.60274 


. 79793 1 


.61658 


.78729 


.63022 


.77641 


56 


5 


.57477 


.81832 


.58896 


.80816 


.60298 


.79776! 


.61681 


.78711 


.63045 


.77623 


55 


6 


.57501 


.81815; 


.58920 


.8079 J 


.60321 


.797581 


.61704 


.78694 


.63068 


.77605 


54 


7 


.57524 


.81798 


.58943 


.80782; 


.60344 


.79741 


.61726 


.78676 


.63090 


.77586 


53 


8 


.57548 


.81782 


.58967 


.807651 


.60367 


.79723! 


.61749 


.78658 


.63113 


.77568 


52 


9 


.57572 


.81765 


.58990 


.80748 


.60390 


.797061 


.61772 


.78640 


.63135 


.77550 


51 


10 


.57596 


.81748 


.59014 


.80730 


.60414 


.79688 


.61795 


.78622 


.63158 


.77531 


50 


11 


.57619 


.81731 


.59037 


.80713 


.60437 


.79671 


.61818 


.78604 


.63180 


.77513 


40 


12 


.57643 


.81714 


.59061 


.80696 


.60460 


.79653 


.61841 


.78586 


.63203 


.77494 


48 


13 


.57667 


.81698 


.59084 


.80679 


.60483 


.79635 


.61864 


.78568 


.63225 


.77476 


47 


14 


.57691 


.81681 


.59108 


.80662 


.60506 


.79618 


.61887 


.78550 


.63248 


.77458 


46 


15 


.57715 


.81664 


.59131 


.80644 


.60529 


.79600 


.61909 


.78532 


.63271 


.77439 


45 


16 


.57738 


.81647 


.59154 


.80627 


.60553 


.79583 


.61932 


.78514 


.63293 


.77421 


44 


17 


.57762 


.81631 


.59178 


.80610 


.60576 


.79565 


.61955 


.78496 


.63316 


.77402 


43 


18 


.57786 


.81614 


.59201 


.80593! 


.60599 


.79547 


.61978 


.78478 


.63338 


.77384 


42 


19 


.57810 


.81597 


.59225 


.80576 


.60622 


.79530 


.62001 


.78460 


.63361 


.77366 


41 


20 


.57833 


.81580 


.59248 


.80558 


.60645 


.79512 


.62024 


.78442 


.63383 


.77347 


40 


21 


.57857 


.81563 


.59272 


.80541' 


.60668 


.79494 


.62046 


.78424 


.63406 


.77329 


39 


22 


.57881 


.81546 


.59295 


.80524! 


.60691 


.79477 


i .62069 


.78405 


.63428 


.77310 


38 


23 


.57904 


.81530 


.59318 


.80507 


.60714 


.79459 


.62092 


.78387 


.63451 


.77292 


37 


24 


.57928 


.81513 


.59342 


.80489 


.60738 


.79441 


.62115 


.78369 


.63473 


.77273 


36 


25 


.57952 


.81496 


.59365 


.80472; 


.60761 


.79424 


.62138 


.78351 


.63496 


.77255 


35 


26 


.57976 


.81479 


.59389 


.80455! 


.60784 


.79406 


.62160 


.78333 


.63518 


.77236 


34 


27 


.57999 


.81462 


.59412 


.80438 


.60807 


.79388 


.62183 


.78315 


63540 


.77218 


33 


28 


.58023 


.81445 


.59436 


.80420; 


.60830 


.79371 


.62206 


.78297 


.63563 


.77199 


32 


29 


.58047 


.81428 


.59459 


.8O403J 


60853 


.79353 


.62229 


.78279 


.63585 


.77181 


31 


30 


.58070 


.81412 


.59482 


.80386| 


.60876 


.79335 


.62251 


.78261 


.63608 


.77162 


30 


31 


.58094 


.81395 


.59506 


.80368! 


.60899 


.79318 


.62274 


.78243 


.63630 


.77144 


29 


32 


.58118 


.81378 


.59529 


.803511 


.60922 


.79300 


.62297 


.78225 


.63653 


.77125 


28 


33 


.58141 


.81361 


.59552 


.80334! 


.60945 


.79282 


.62320 


.78206 


.63675 


.77107 


27 


34 


.58165 


.81344 


.59576 


.80316' 


.60968 


.79264 


.62342 


.78188 


.63698 


.77088 


26 


35 


.58189 


.81327 


.59599 


.80299 


.60991 


.79247 


.62365 


.78170 


.63720 


.77070 


25 


36 


.58212 


.81310 


.59622 


.80282! 


.61015 


.79229 


.62388 


.78152 


.63742 


.77051 


24 


37 


.58236 


.81293 


.59646 


.80264! 


.61038 


.79211 


.62411 


.78134 


.63765 


.77033 


23 


38 


.58260 


.81276 


.59669 


.80247; 


.61061 


.79193 


.62433 


.78116 


.63787 


.77014 


22 


39 


.58283 


.81259 


.59693 


.80230' 


.61084 


.79176 


.62456 


.78098 


.63810 


.76996 


21 


40 


.58307 


.81242 


.59716 


.80212! 


.61107 


.79158 


.62479 


.78079 


.63832 


.76977 


20 


41 


.58330 


.81225 


.59739 


.80195 


.61130 


.79140 


.62502 


.78061 


.63854 


.76959 


19 


42 


.58354 


.81208 


.59763 


.801781 


.61153 


.79122 


.62524 


.78043 


.63877 


.76940 


18 


43 


.58378 


.81191 


.59786 


.801601 


.61176 


.79105 


.62547 


.78025 


.63899 


.76921 


17 


44 


.58401 


.81174 


.59809 


.80143' 


.61199 


.79C87 


.62570 


.780071 


.63922 


.76903 


16 


45 


.58425 


.81157 


.59832 


.80125 ! 


.61222 


.79069 


.62592 


.77988 


.63944 


.76884 


15 


46 


.58449 


.81140 


.59856 


.80108 


.61245 


.79051 


.62615 


.77970' 


.63966 


.76866 


14 


47 


.58472 


.81123; 


.59879 


.80091 ! 


.61268 


.79033 


.62638 


.77952! 


.63989 


.76847 


13 


48 


.58496 


.81106J 


.59902 


.80073 


.61291 


.79016 


.62660 


.77934 


.64011 


.76828 


12 


49 


.58519 


.81089 


.59926 


.80056 


.61314 


.78998 


.62683 


.77916^ 


.64033 


.76810 


11 


50 


.58543 


.81072 


.59949 


.80038 


.61337 


.78980 


.62706 


.77897; 


.64056 


.76791 


10 


51 


.58567 


.81055 


.59972 


. 80021 ! 


.61360 


.78962 


.62728 


.77879 


.64078 


.76772 


9 


52 


.58590 


.810381 


.59995 


80003 


.61383 


.78944 


.62751 


.77861 


.64100 


.76754 


8 


53 


.58614 


.810211 


.60019 


.79986 


.61406 


.78926 


.62774 


.77843 


.64123 


.76735 


7 


54 


.58637 


.810041 


.60042 


.79968 


.61429 


.78908 


.62796 


.77824 


.64145 


.76717 


6 


55 


.58661 


.80987 | 


.60065 


.79951 


.61451 


.78891 


.62819 


. 77806 i 


.64167 


.76698 


5 


56 


.58684 


.80970 


.60089 


.799:34 


.61474 


.78873 


.62842 


.77788; 


.64190 


.76679 


4 


57 


.58708 


.80953 


.60112 


.79916 


.61497 


.78855 


.62864 


.77769! 


.64212 


.76661 


3 


58 


.58731 


.80936' 


.60135 


.79899 


.61520 


.78837 


.62887 


.77751! 


.64234 


.76642 


2 


59 


.58755 


.80919 


.60158 


.79881 


.61543 


.78819 


.62909 


.77733 


.64256 


.76623 


1 


60 

/ 


.58779 
Cosin 


.80902 
Sine 


.60182 
Cosin 


.79864 
Sine 


.61566 


.78801 


.62932 
Cosin 


.77715 
Sine 


.64279 


.76604 




/ 


Cosin 


Sine 


Cosin 


Sine 


54° 


53° 


62° 


61- 


50° 



208 



NATURAL SINES AND COSINES. 



t 

"o 


40° 


41° 


42° 


43° 


44° 


> 

60 


Sine 


Cosin 
.76604 


Sine 
.65606 


Cosin 
.75471 


Sine 


Cosin 


Sine 
768200 


Cosin 
.73135 


Sine Cosin 


.64279 


.66913 


.74314 


.69466 .71934 


1 


.64301 


.76586! 


.65628 


.75452 


.66935 


.74295 


.68221 


.73116 


.694871.71914 


59 


2 


.64323 


.76567 


.65650 


.75433 


.66956 


.74276 


.68242 


.73096 


.69503 .71894 


58 


3 


.64346 


.76548 


.65672 


.75414 


.66978 


.74256 


.68264 


.73076 


.69529 .71873 


57 


4 


.64368 


.76530 


.65694 


.75395 


.66999 


.74237 


.68285 


.73056 


.69549 .71853 


56 


5 


.64390 


.76511 


.65716 


. 75375 


.67021 


.74217 


.68306 


.73036 


.69570 .71833 


55 


6 


.64412 


.76492 


.65738 


.75356 


.67043 


.74198 


.68327 


.73016 


.69591 .71813 


54 


7 


.64435 


.76473 


.65759 


.75337 


.67064 


.74178 


.68349 


.72996 


.69612 .71792 


53 


8 


.64457 


. 76455 ! 


.65781 


.75318 


.67086 


.74159 


.68370 


.72976 


.69633 .71772 


52 


9 


.64479 


.76435 


.65803 


.75299 


.67107 


.74139 


.68391 


.72957 


.69654 


.71752 


51 


10 


.64501 


.76417 


.65825 


.75280 


.67129 


.74120 


.68412 


.72937 


.69675 


.71732 


50 


11 


.64524 


.76398' 


.65847 


.75261 


.67151 


.74100 


.68434 


.72917 


.69696 


.71711 


49 


12 


.64546 


.76380; 


.65869 


.75241 


.67172 


.74080 


.68455 


.72897 


.69717 


.71691 


48 


13 


.64568 


.76361, 


.65891 


.75222 


.67194 


.74061 


.68476 


. 72877 


.69737 


.71671 


47 


14 


.64590 


. 76342 


.65913 


.75203: 


.67215 


.74041 


.68497 


.72857 


.69758 .71650 


46 


15 


.64612 


.76323 


.65935 


.75184 


.67237 


.74022 


.68518 


.72837 


.69779 .71630 


45 


16 


.64635 


.76304 


.65956 


.751651 


.67258 


.74002 


.68539 


.72817 


.69800 


.71610 


44 


17 


.64657 


.76236 


.65978 


.75146; 


.67280 


.73983 


.68561 


.72797 


.69821 


.71590 


43 


18 


.64679 


.7625? 


.63000 


.75128 ' 


.67301 


.73963 


.68582 


.72777 


.69842 


.71569 


42 


19 


.64701 


.76243 


.65022 


.75107: 


.67323 


.73944 


.68603 


.72757 


.69862 


.71549 


41 


20 


.64723 


.76229 


.66044 


.75083 


.67344 


.73924 


.68624 


.72737 


.69883 


.71529 


40 


21 


.64746 


. 76210 ' 


.66066 


.75039 


.67366 


.73904 


.68645 


.72717 


.69904 


.71508 


39 


22 


.64768 


.76192 


.65033 


.75950' 


.67337 


.738S5 


.63666 


.72697 


.69925 


.71488 


38 


23 


.64790 


.76173 


.65109 


.75030 


.67409 


.73835 


.63633 


.72677 


.69946 


.71468 


37 


24 


.64812 


.76154 


.65131 


.75311: 


.67430 


. 73346 


.68709 


.72657; 


.69966 


.71447 


36 


25 


.64834 


.76135 


.65153 


.74902: 


.67452 


.73823 


.68730 


.72637! 


.69987 


.71427 


35 


26 


.64856 


.76116 


.65175 


.74073 


.67473 


.73803 


.63751 


.72617! 


.70008 


.71407 


34 


27 


.64878 


.76097 


.65197 


.74953 


.67495 


.73787 


.68772 


. 72597 i 


.70029 


.71386 


33 


28 


.64901 


.76078 


.65218 


. 74934 ' 


.67516 


.73767 


.63793 


. 72577 j 


.70049 


.71366 


32 


29 


.64923 


.76059 


.65240 


.74915 


.67538 


.73747 


.68814 


. 72557 


.70070 


.71345 


31 


30 


.64945 


.76041 


.66252 


.74896 


.67559 


.73728 


.68835 


.72537 


.70091 


.71325 


30 


31 


.64967 


. 76022 ' 


.66284 


.74876 


.67580 


.73703 


.68857 


.72517 


.70112 


.71305 


29 


32 


.64989 


.76003 


.65306 


.74357 


.67632 


.73683 


.68878 


. 72497 ■ 


70132 


.71284 


28 


33 


.65011 


.75934 


.65327 


.74333 


.67623 


.73639 


.68899 


.72477| 


.70153 


.71264 


27 


34 


.65033 


. 75985 ; 


.65349 


.74818 


.67645 


.73649 


.68920 


.72457 


.70174 


.71243 


26 


35 


. 65055 


.75946 


.65371 


.74799 


.67666 


.73629 


.68941 


.72437 


.70195 


.71223 


35 


36 


.65077 


.75927 


.65393 


.74780 


.67688 


.73610 


.68962 


.72417 


.70215 


.71203 


24 


37 


.65100 


.75903 


.66414 


.74769 


.67709 


.73590 


.68983 


.72397 


.70236 


.71182 


23 


38 


.65122 


.75889 


.65438 


.74741 


.67730 


73570 


.69004 


.72377 


.70257 


.71162 


22 


39 


.65144 


.75870 


.65458 


.74722 


.67752 


.73551 


.69025 


.72357 


.70277 


.71141 


21 


40 


.65166 


.75851; 


.66480 


.74703 


.67773 


.73531 


.69046 


.72337 


.70298 


.71121 


20 


41 


.65188 


.75832 


.66501 


.74683 


.67795 


.73511 


.69067 


.72317 


.70319 


.71100 


19 


42 


.65210 


.75813 


.65523 


.74684: 


.67816 


.73491 


.69088 


.72297 


.70339 


.71080 


18 


43 


.65232 


.75794 


.65545 


.74644 


.67837 


.73472 


.69109 


.72277 


.70360 


.71059 


17 


44 


.65254 


.75775 


.68566 


.74625 


.67859 


.73452 


.69130 


.72257 


.70381 


.71039 


16 


45 


.65276 


.75756 


.66588 


.74603 


.67880 


.73432 


.69151 


.72236 


.70401 


.71019 


15 


46 


.65298 


.757381 


.68610 


.74588 


.67901 


.73413 


.69172 


.72216 


.70422 


.70998 


14 


47 


.65320 


.75719 


. 68532 


.74567 


.67923 


.73393 


.69193 


.72196 


.70443 


.70978 


13 


48 


.65342 


.75700 


.66653 


.74548 


.67944 


.73373 


.69214 


.72176 


.70463 


.70957 


12 


49 


.65364 


.75630 


.66675 


.74528 


.67965 


.73353 


.69235 


.72156 


.70484 


.70937 


11 


50 


.65336 


.75661) 


.66697 


.74509: 


.67987 


.73333 


.69256 


.72136 


.70505 


.70916 


10 


51 


.65408 


.75642: 


.66718 


.74489 


.68008 


.73314 


.69277 


.72116 


.70525 


.70896 


9 


52 


.65430 


.75623 


.66740 


.74470 


.68029 


.73294 


.69298 


.72095 


.70546 


.70875 


8 


53 


.65452 


.75604 


.66762 


.74451 


i .68051 


.73274 


; .69319 


.72075 


.70567 


.70855 


7 


54 


.65474 


.75585 


.66783 


.74431 


i .68072 


.732.54 


i .69340 


.72055 


.70587 


.70834 


6 


55 


.65496 


. 75566 


.66805 


.74412 


.68093 


.73234 


; .69361 


.72035 


.70608 .70813 


5 


56 


.65518 


. 75547 


.66827 


.74392 


.68115 


.73215 


i .69382 


.72015 


.70628 .70793 


4 


57 


.65540 


.75528 


.66848 


.743731 


.68136 


.73195 


\ .69403 


.71995 


.70649 .70772 


3 


58 


.65562 


.75509 


.66870 


.74353 


.68157 


.73175 


i .69424 


.71674 


.70670 .70752 


2 


59 


.65584 


.75490 


.66891 


.74334 


.68179 


.73155 


! .69445 


.71954 


.70690 .70731 


1 


60 

/ 


.65606 


.75471 i 


.66913 


.74314 


.68200 


.73135 


i .69466 


. 71934 ' 


I .70711 .70711 
Cosin j Sine 




f 


Cosin j Sine 


Cosin 


Sine 


Cosin 


Sine 


! Cosin 


Sine j 


49° 


1 48° 


47° 


1 46° 


45° 



209 



NATURAL TANGENTS AND COTANGENTS. 



/ 

"o 


0° 


1° 


2° 


S 


• 


/ 

60 


Tang 


Cotang 


Tang 
.01746 


Cotang 


Tang 

.03492 


Cotang 


Tang 
.05241 


Cotang 


.00000 


Infinite. 


57.2900 


28.6863 


19.0811 


X 


.00029 


3437.75 


.01775 


56.3506 


.03521 


28.3994 


.05270 


18.9755 


59 


2 


.00058 


1718.87 


.01804 


55.4415 


.03550 


28.1664 


.05299 


18.8711 


58 


3 


.00087 


1145.92 


.01833 


54.5613 


.03579 


27.9372 


.05328 


13.7678 


57 


4 


.00116 


859.436 


.01862 


53.70S6 


.03609 


27.7117 


.05357 


18.6656 


56 


5 


.00145 


6S7.549 


.01691 


52.8821 


.03638 


27.4899 


.05387 


18.5645 


55 


C 


.00175 


572.957 


.01 920 


52.0807 


.03667 


27.2715 


.05416 


18.4645 


54 


7 


.00204 


491.106 


.01949 


51.3032 


.03696 


27.0566 


.05-445 


18.3655 


53 


8 


.00233 


429.718 


.01978 


50.5485 | 


.03725 


26.8450 


.05474 


18.2677 


52 


9 


.00262 


381.971 


.02007 


49.8157 ! 


.03754 


26.6367 


.05503 


18.1708 


51 


10 


.00291 


343.774 


.02036 


49.1039 


.03783 


26.4316 


.05533 


18.0750 


50 


11 


.00320 


312.521 


.02033 


48.4121 


.03S13 


23.2296 


.05562 


17.9802 


49 


12 


.003-19 


286.478 


.02095 


47.7395 


.03342 


26.0307 


.05591 


17.8863 


48 


13 


.00378 


264.441 


.02124 


47.0853 


.03871 


25.8343 


.05620 


17.7934 


47 


14 


.00407 


245.552 


.02153 


46.4489 


.03300 


25.6418 


.05649 


17.7015 


46 


15 


.00436 


229.182 


.02182 


45.8294 


.03929 


25.4517 


.05078 


17.6106 


45 


16 


.00465 


214.858 


.02211 


45.2261 


.03958 


25.2644 


.05708 


17.5205 


44 


17 


.00495 


202.219 


.02240 


44.6386 


.03987 


25.0798 


.05737 


17.4314 


43 


18 


.00524 


190.984 


.02269 


44.0661 


.04016 


24.8978 


.05766 


17.3432 


42 


19 


.00553 


180.932 


.02293 


43.5081 


.04046 


24.71S5 


.05795 


17.2558 


41 


20 


.00582 


171.885 


.02328 


42.9641 


.04075 


24.5418 


.05824 


17.1693 


40 


21 


.00611 


163.700 


.02357 


42.4335 


.04104 


24.3675 


.05854 


17.0837 


39 


22 


.00040 


156.259 


.02386 


41.9158 


.04133 


24.1957 


.05883 


16.9990 


38 


23 


.00669 


149.465 


.02415 


41.4106 


.04162 


24.0263 


.05912 


16.9150 


37 


24 


.00698 


143.237 


.02444 


40.9174 


.04191 


23.8593 


.05941 


16.8319 


36 


25 


.00727 


137.507 


.02473 


40.4358 


.04220 


23.6945 


.05970 


16.7496 


35 


26 


.00756 


132.219 


.02502 


39.9655 


.04250 


23.5321 


.05999 


16.6681 


34 


27 


.00785 


127.321 


.02531 


39.5059 


.04279 


23.3718 


.06029 


16.5874 


33 


28 


.00815 


122.774 


.02560 


39.0568 


.04308 


23.2137 


.06058 


16.5075 


32 


29 


.00844 


118.540 


.02589 


38.6177 


.04337 


23.0577 


.06087 


16.4283 


31 


30 


.00873 


114.589 


.02619 


38.1885 


.04366 


22.9038 


.06116 


16.3499 


30 


31 


.00902 


110.892 


.02648 


37.7686 


.04395 


22.7519 


.06145 


16.2722 


29 


32 


.00931 


107.426 


.02677 


37.3579 


.04424 


22.6020 


.06175 


16.1952 


23 


33 


.00960 


104.171 


.02706 


36.9560 


.04454 


22.4541 


.06204 


16.1190 


27 


34 


.00989 


101.107 


.02735 


36.5627 


.04483 


22.3081 


.06233 


16.0435 


26 


35 


.01018 


98.2179 


.02764 


36.1776 


.04512 


22.1640 


.06262 


15.9687 


25 


36 


.01047 


95.4895 


.02793 


35.8006 


.04541 


22.0217 


.06291 


15.8945 


24 


37 


.01076 


92.9085 


.02822 


35.4313 


.04570 


21.8813 


.06321 


15.8211 


23 


38 


.01105 


90.4633 


.02851 


35.0695 


.04599 


21.7426 


.06350 


15.7483 


22 


39 


.01135 


88.1436 


.02881 


34.7151 


.04628 


21.6056 


.06379 


15.6762 


21 


40 


.01164 


85.9398 


.02910 


34.3678 


.04658 


21.4704 


.06408 


15.6048 


20 


41 


.01193 


83.8435 


.02939 


34.0273 


.04687 


21.3369 


.06437 


15.5340 


19 


42 


.01222 


81.8470 


.02968 


33.6935 


.04716 


21.2049 


.06467 


15.4638 


18 


43 


.01251 


79.9434 


.02997 


33.3662 


.04745 


21.0747 


.06496 


15.3943 


17 


44 


.01280 


78.1263 


.03026 


33.0452 


.04774 


20.9460 


.06525 


15.3254 


16 


45 


.01309 


76.3900 


.03055 


32.7303 


.04803 


20.8188 


.06554 


15.2571 


15 


46 


.C1338 


74.7292 


.03084 


32.4213 


.04833 


20.6932 


.06584 


15.1893 


14 


47 


.01367 


73.1390 


.03114 


32.1181 


.04862 


20.5691 


.06613 


15.1222 


13 


48 


.01396 


71.6151 


.03143 


31.8205 ! 


.04891 


20.4465 


.06642 


15.0557 


12 


49 


.01425 


70.1533 


.03172 


31.5284 i 


.04920 


20.3253 


.06671 


14.9898 


11 


50 


.01455 


68.7501 


.03201 


31.2416 


.04949 


20.2056 


.06700 


14.9244 


10 


51 


.01484 


67.4019 


.03230 


30.9599 


.04978 


20.0872 


.06730 


14.8596 


9 


52 


.01513 


66.1055 


.03259 


30.6833 ! 


.05007 


19.9702 


.06759 


14.7954 


8 


53 


.01542 


64.8580 


.03288 


30.4116 


.05037 


19.8546 


.06788 


14.7317 


7 


54 


.01571 


63.6567 


.03317 


30.1446 


.05066 


19.7403 


.06817 


14.6685 


6 


55 


.01600 


62.4992 


.03346 


29.8823 


.05095 


19.6273 


.06847 


14.6059 


5 


56 


.01629 


61.3829 


.03376 


29.6245 


.05124 


19.5156 


.06876 


14.5438 


4 


57 


.01658 


60.3058 


.03405 


29.37111 


.05153 


19.4051 


.06905 


14.4823 


3 


58 


.01687 


59.2659 


.03434 


29.1220 


.05182 


19.2959 


.06934 


14.4212 


2 


59 


.01716 


58.2612 


.03463 


28.8771 


.05212 


19.1879 


.06963 


14.3607 


1 


60 
/ 


.01746 


57.2900 


.03492 


28.6363 


.05241 


19.0811 


.06993 


14.3007 





Cotang 


Tang 


■Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


89° 


88° 


87° 


8 


6° 



210 



NATURAL TANGENTS AND COTANGENTS , 






4° 


5° 


6° 


7° 


60 


Tang Cotang 
.06993 14.3007 


Tang 
.08749" 


Cotang 


Tang Cotang 


Tang 


Cotang 


11.4301 


.10510 9.51436 


.12278 


8.14435 


1 


.07022 14.2411 


.08778 


11.3919 


.10540 9.48781 


.12303 


8.12481 


59 


2 


.07051 i 14.1821 


.08807 


11.3540 


.10569 i 9.46141 ! 


.12333 


8.10530 


58 


3 


.07U80 14.1235 


.08837 


11.3163 


.10599 '■ 9.43515 


.12367 


8.086C0 


57 


4 


.07110 


14.0655 


.08866 


11.2789 


.10628 9.40904 


.12397 


8.C6674 


56 


5 


.07139 


14.0079 


.08895 


11.2417 


.10057 


9.38307 ; 


.12426 


8.04756 


55 


6 


.07168 


13.9507 i 


.08925 


11.2048 


.10687 


9.85724 


.12456 


8.02848 


54 


7 


.07197 


13.8940 


.08954 


11.1681 


.10716 


9 . 33155 


.12485 


8.00948 


53 


8 


.07227 


13.8378 


.08983 


11.1316 


.10746 


9.30599 i 


.12515 


7.99058 


52 


9 


.07256 


13.7821 | 


.09013 


11.0954 


.10775 


9.28058 


.12544 


7.97176 


51 


10 


.07285 


13.7267 1 


.09042 


11.0594 


.10805 


9.25530 


.12574 


7.95302 


50 


11 


.07314 


13.6719 ! 


.09071 


11.0237 


.10834 


9.23016 ' 


.12603 


7.93488 


49 


12 


.07344 13.0174 


.09101 


10.9882 


.10863 


9.20516 


.12633 


7.91582 


48 


13 


.07373 13.5634 : 


.09130 


10.9529 


.10803 


9.18028 


.12662 


7.89734 


47 


14 


.07402 


13.5098 


.09159 


10.9178 


.10922 


9.15554 


.12692 


7.87895 


46 


15 


.07431 


13.4566 i 


.09189 


10.8829 


.10952 


9.13093 


.12722 


7.86064 


45 


10 


.07461 


13.4039 


.09218 


10.8483 


.10981 


9.1C646 


.12751 


7.84242 


44 


17 


.07490 


13.3515 ! 


.09247 


10.8139 


.11011 


9.08211 


.12781 


7.82428 


43 


18 


.07519 


13.2996 ! 


.09277 


10.7797 


.11040 


9.05789 


.12810 


7.80622 


42 


19 


.07543 


13.2480 ; 


.09806 


10.7457 


.11070 


9.03379 


' .12840 


7.78825 


41 


20 


.07578 


13.1969 i 


.09335 


10.7119 


.11099 


9.00963 


.12869 


7.77035 


40 


21 


.07607 


13.1461 


.09365 


10.0783 


.11128 


8.98598 ! 


' .12899 


7.75254 


39 


22 


.07636 


13.0958 


.09394 


10.0450 


.11158 


8.96227 


1 .12929 


7.73480 


38 


23 


.07065 


13.0458 


.09423 


10.6118 


.11187 


8.93867 


i .12958 


7.71715 


37 


24 


.07695 


12.9962 


.C9453 


10.5789 


.11217 


8.91520 


.12988 


7.69957 


36 


25 


.07724 


12.9469 


.09482 


10.5462 


.11246 


8.89185 


1 .13017 


7.68208 


35 


26 


.07753 


12.8981 


.09511 


10.5136 


.11276 


8.86862 


! .13047 


7.66466 


34 


27 


.07782 


12.8496 


.C9541 


10.4813 


.11805 


8.84551 


' .13076 


7.64732 


33 


28 


.07812 


12.8014 


.09570 


10.4491 


.11835 


8.82252 


1 .13106 


7.63005 


32 


29 


.07S41 


12.7536 


.09000 


10.4172 


.11864 


8.79964 


.13136 


7.61287 


31 


30 


.07870 


12.7062 


.09629 


10.3854 


.11394 


8.77689 


.13165 


7.59575 


30 


31 


.07899 


12.6591 


.09658 


10.3538 


.11423 


8.75425 


.13195 


7.57872 


29 


32 


.07929 


12.6124 


.09688 


10.3224 


.11452 


8.73172 


.13224 


7.56176 


28 


33 


.07958 12.5060 I 


.09717 


10.2913 


.11482 


8.70931 


.13254 


7.54487 


27 


34 


.07987 


12.5199 ! 


.09746 


10.2602 


.11511 


8.68701 


.13284 


7.52806 


26 


85 


.08017 


12.4742 i 


.09776 


10.2294 


.11541 


8.66482 


.13313 


7.51132 


25 


86 


.08046 


12.4288 


.09805 


10.1988 


.11570 


8.64275 


! .13343 


7.49465 


24 


37 


.0SC75 


12.3838 


.09834 


10.1683 


i .11600 


8.62078 


.13372 


7.47806 


23 


88 


.0S104 


12.3390 


.09664 


10.1881 


i .11629 


8.59893 


.13402 


7.46154 


22 


39 


.08134 


12.2946 


.09893 


10.1080 


i .llf ' 


8.57718 


i .13432 


7.44509 


21 


40 


.08163 


12.2505 


.09923 


10.0780 


.11688 


8.55555 


.13461 


7.42871 


20 


41 


.08192 


12.2067 


.09952 


10.0483 


.11718 


i 53402 


.13491 


7.41240 


l l 


42 


.08221 


12.1632 : 


.09981 


10.0187 


.11747 


8.51259 


.13521 


7.39616 


18 


43 


.08251 


12.1201 


.10011 


9.98931 


.11777 


8.49128 


.13550 


7.37999 


*I 


44 


.08280 


12.0772 


.10040 


9.96007 


, .11806 


8.47007 


.13580 


7.36389 


16 


45 


.08809 


12.0346 


.10069 


9.93101 


i .11836 


8.44896 


.13609 


7.34786 


15 


46 


.08339 


11.9923 


.10099 


9.90211 


1 .11865 


8.42795 


.13639 


7.33190 


14 


47 


.08368 


11.9504 


.10128 


9.87338 


j .11895 


8.40705 


.13669 


7.31600 


13 


48 


.08397 


11.9087 


.10158 


9.84482 


.11924 


8.38625 


.13698 


7.30018 


12 


49 


.08427 


11.8673 


.10187 


9.81641 


.11954 


8.36555 


: .13728 


7.28442 


11 


50 


.08456 


11.8262 


.10216 


9.78817 


.11983 


8.34496 


.13758 


7.26873 


10 


51 


.08485 


11.7853 


; .10246 


8.76009 


.12013 


8.32446 


.13787 


7.25310 


9 


52 


.08514 


11.7448 


.10275 


9.73217 1 


.12042 


8.30406 


i .13817 


7.23754 


8 


53 


.08544 


11.7045 


! .10305 


9.70441 ■ 


.12072 


8.28376 


i .13846 


7.22204 


7 


54 


.0S573 


11.6645 . 


! .103:34 


9.67680 i 


.12101 


8.26355 


i .13876 


7.20661 


6 


55 


.08602 


11.6248 


' .10363 


9.64935 : 


.12131 


8.24345 


.13906 


7.19125 


5 


56 


.08632 


11.5853 


1 .10393 


9.62205 1 


.12160 


8.22344 


.13935 


7.17594 


4 


57 


.08661 


11.5461 


.10422 


9.59490 ; 


.12190 


8.20352 


.13965 


7.16071 


3 


58 


.08690 i 11.5072 


: .10452 


9.56791 


.12219 


8.18370 


.13995 


7.14553 


2 


59 


.08720 11.4685 


, .10481 


9.54106 


.12249 


8.16398 


.14024 


7.13042 


1 


60 

/ 


.08749 1 11.4301 


1 .10510 


9.51436 


.12278 8.14435 


.14054 


7.11537 



/ 


Cotang ! Tang 


Cotang 


Tang 


Cotang Tang 


Cotang 


Tang 


85° 


84° 


83° 


82° 



211 



NATUBAL TANGENTS AND COTANGENTS. 



t 

~0 


8° 


9° 


10° 


11° 


/ 
60 


Tang 
.14054 


Cotang 


Tang 

.15838 


Cotang 


Tang 
.17633 


Cotang 


Tang 
.19438 


Cotang 


7.11537 


6.31375 


5.67128 


5.14455 


1 


.14084 


7.10038 


.15b08 


6.30189 


.17663 


5.66165 


.19468 


5.13658 


59 


2 


.14113 


7.08546 


.15898 


6.29007 


.17693 


5.65205 


.19498 


5.12862 


58 


3 


.14143 


7.07059 


.15928 


6.27829 


.17723 


5.64248 


.19529 


5.12069 


57 


4 


.14173 


7.05579 


.15958 


6.26655 


.17753 


5.63295 


.19559 


5.11279 


56 


5 


.14202 


7.04105 


.15938 


6.25486 


.17783 


5.62344 


.19589 


5.10490 


55 


6 


.14232 


7.02637 


.16017 


6.24321 


.17813 


5.61397 


.19619 


5.09704 


54 


7 


.14262 


6.91174 


.16047 


6.23160 


.17843 


5.60452 


.19649 


5.08921 


53 


8 


.14291 


6.99718 


.16077 


6.22003 


.17873 


5.59511 


.19680 


5.08139 


52 


9 


.14321 


6.98268 


.16107 


6.20851 


.17903 


5.58573 


.19710 


5.07360 


51 


10 


.14351 


6.96823 


.16137 


6.19703 


.17933 


5.57638 


.19740 


5.06584 


50 


11 


.14381 


6.95385 


.16167 


6.18559 


.17963 


5.56706 


.19770 


5.05809 


49 


12 


.14410 


6.93952 


.16196 


6.17419 


.17993 


5.55777 


.19801 


5.05037 


48 


13 


.14440 


6.92525 


.16226 


6.16283 


.18023 


5.54851 


.19831 


5.042:.7 


47 


U 


.14470 


6.91104 


.16256 


6.15151 


.18053 


5.53927 


.19861 


5.03499 


46 


15 


.14499 


6.89688 


.16286 


6.14023 


.18083 


5.53007 


.19891 


5.02734 


45 


16 


.14529 


6.88278 


.16316 


6.12899 


.18113 


5.52090 


.19921 


5.01971 


44 


17 


.14559 


6.86874 


.16346 


6.11779 


.18143 


5.51176 


.19952 


5.01210 


43 


18 


.14588 


6.85475 


.16376 


6.10664 


.18173 


5.50264 


.19982 


5.00451 


42 


19 


.14618 


6.84082 


.16405 


6.09552 


.18203 


5.49356 


.20012 


4.99695 


41 


20 


.14648 


6.82694 


.16435 


6.08444 


.18233 


5.48451 


.20042 


4.98940 


40 


21 


.14678 


6.81312 


.16465 


6.07340 


.18263 


5.47548 


.20073 


4.98188 


39 


22 


.14707 


6.79936 


.16495 


6.06240 


.18293 


5.46648 


.20103 


4.97438 


38 


23 


.14737 


6.78564 


.16525 


6.05143 


.18323 


5.45751 


.20133 


4.96690 


37 


24 


.14767 


6.77199 


.16555 


6.04051 


.18353 


5.44857 


.20164 


4.95945 


36 


25 


.14796 


6.75838 


.16585 


6.02962 


.18384 


5.43966 


.20194 


4.95201 


35 


26 


.14826 


6.74483 


.16615 


6.01878 


.18414 


5.43077 


.20224 


4.94460 


34 


27 


.14856 


6.73133 


.16645 


6.00797 


.18444 


5.42192 


.20254 


4.93721 


33 


28 


.14886 


6.71789 


.16674 


5.99720 


.18474 


5.41309 


.20285 


4.92984 


32 


29 


.14915 


6.70450 


.16704 


5.9S646 


.18504 


5.40429 


.20315 


4.92249 


31 


30 


.14945 


6.69116 


.16734 


5.97576 


.18534 


5.39552 


.20345 


4.91516 


30 


31 


.14975 


6.67787 


.16764 


5.96510 


.18564 


5.38677 


.20376 


4.90785 


29 


32 


.15005 


6.66463 


.16794 


5.95448 


.18594 


5.37805 


.20406 


4.90056 


28 


33 


.15034 


6.65144 


.16824 


5.94390 


.18624 


5.36936 


.20436 


4.89&30 


27 


34 


.15064 


6.63831 


.16854 


5.93335 


.18654 


5.36070 


.20466 


4.88605 


26 


35 


.15094 


6.62523 


.16884 


5.92283 


.18684 


5.35206 


.20497 


4.878S2 


25 


36 


.15124 


6.61219 


.16914 


5.91236 


.18714 


5.34345 


.20527 


4.87162 


24 


37 


.15153 


6.59921 


.16944 


5.90191 


.18745 


5.33487 


.20557 


4.86444 


23 


38 


.15183 


6.58627 


.16974 


5.89151 


.18775 


5.32631 


.20588 


4.85727 


22 


39 


.15213 


6.57339 


.17004 


5.83114 


.18805 


5.31778 


.20618 


4.85013 


21 


40 


.15243 


6.56055 


.17033 


5.87080 


.18835 


5.30928 


.20648 


4.84300 


20 


41 


.15272 


6.54777 


.17063 


5.86051 


.18865 


5.30080 


.20679 


4.83590 


19 


42 


.15302 


6.53503 


.17093 


5.85024 


.18895 


5.29235 


.20709 


4.82882 


18 


43 


.15332 


6.52234 


.17123 


5.84001 


.18925 


5.28393 


.20739 


4.82175 


17 


44 


.15362 


6.50970 


.17153 


5.82982 


.18955 


5.27553 


.20770 


4.81471 


16 


45 


.15391 


6.49710 


.17183 


5.81966 


.18986 


5.26715 


.20800 


4.80769 


15 


46 


.15421 


6.48456 


.17213 


5.80953 


.19016 


5.25880 


.20830 


4.80068 


14 


47 


.15451 


6.47206 


.17243 


5.79944 


.19046 


5.25048 


.20861 


4.79370 


13 


48 


.15481 


6.45961 


.17273 


5.78938 


.19076 


5.24218 


.20891 


4.78673 


12 


49 


.15511 


6.44720 


.17303 


5.77936 


.19106 


5.23391 


.20921 


4.77978 


11 


50 


.15540 


6.43484 


.17333 


5.76937 


.19136 


5.22566 


.20952 


4.7?286 


10 


51 


.15570 


6.42253 


.17363 


5.75941 


.19166 


5.21744 


.20982 


4.76595 


9 


52 


.15600 


6.41026 


.17393 


5.74949 


.19197 


5.20925 


.21013 


4.75906 


8 


53 


.15630 


6.39804 


.17423 


5.73980 


.19227 


5.20107 


.21043 


4.75219 


7 


54 


.15660 


6.38587 


.17453 


5.72974 


.19257 


5.19293 


.21073 


4.74534 


6 


55 


.15689 


6.37374 


.17483 


5.71992 


.19287 


5.18480 


.21104 


4.73851 


5 


56 


.15719 


6.36165 


.17513 


5.71013 


.19317 


5.17671 


.21134 


4.73170 


4 


57 


.15749 


6.34961 


.17543 


5.70037 


.19347 


5.16863 


.21164 


4.72490 


3 


58 


.15779 


6.33761 


.17573 


5.69064 


.19378 


5.16058 


.21195 


4.71813 


2 


59 


.15809 


6.32566 


.17603 


5.68094 


.19408 


5.15256 


.21225 


4.71137 


1 


60 


.15838 


6.31375 


.17633 


5.67128 


.19138 


5.14455 


.21256 

Cotang 


4.70463 


_0 

/ 


/ 


Cotang 


Tang 


Cotang | Tang 


Cotang 


Tang 


Tang 


81° 


80° 


79° 


78° 



•212 



NA TUBAL TANGENTS AND COTANGENTS. 



/ 
~0 


12 b 


13° 


14° 


15° 


60 


Tang 


Cotang 


Tang 

.23087 


Cotang 
4.33148 


Tang 
.24933 


Cotang ! 


Tang | Cotang 


.21256 


4.70463 


4.01078 1 


.26795 


3.73205 


1 


.21286 


4.69791 


.23117 


4.32573 


.24964 


4.00582 


.26826 


3.72771 


59 


2 


.21316 


4.69121 


.23148 


4.32001 


.24995 


4.00086 


.26857 


3.72338 


58 


3 


.21347 


4.68452 


.23179 


4.31430 


.25026 


3.99592 


.26888 


3.71907 


57 


4 


.21377 


4.67786 


.23209 


4.30860 


.25056 


3.99099 


.26920 


3.71476 


56 


5 


.21408 


4.67121 


.23240 


4.30291 


.25087 


3.98607 


.26951 


3.71046 


55 


6 


.21438 


4.66458 


.23271 


4.29724 


.25118 


3.98117 


.26982 


3.70616 


54 


7 


.21469 


4.65797 


.23301 


4.29159 


.25149 


3.97627 


.27013 


3.70188 


53 


8 


.21499 


4.65138 


.23332 


4.28595 


.25180 


3.97139 


.27044 


3.69761 


52 


9 


.215£9 


4.64480 


.23363 


4.28032 


.25211 


3.96651 


.27076 


3.69335 


51 


10 


.21560 


4.63825 


.23393 


4.27471 


.25242 


3.96165 


.27107 


3.68909 


50 


11 


.21590 


4.63171 


.23424 


4.26911 


.25273 


3.95680 


.27138 


3.68485 


49 


12 


.21621 


4.62518 


.23455 


4.26352 


.25304 


3.95196 


.27169 


3.68061 


48 


13 


.21651 


4.61868 


.23485 


4.25795 


.25335 


3.94713 


.27201 


3.67638 


47 


14 


.21682 


4.61219 


.23516 


4.25239 


.25366 


3.94232 


.27232 


3.67217 


46 


15 


.21712 


4.60572 


.23547 


4.24685 


.25397 


3.93751 


.27263 


3.66796 


45 


16 


.21743 


4.59927 


.23578 


4.24132 


.25428 


3.93271 


.27294 


3.66376 


44 


17 


.21773 


4.59283 


.23608 


4.23580 


.25459 


3.92793 


.27326 


3.65957 


43 


18 


.21804 


4.58641 


.23639 


4.23030 


.25490 


3.92316 


.27357 


3.65538 


42 


19 


.21834 


4.58001 


.23670 


4.22481 


.25521 


3.91839 


.27388 


3.65121 


41 


20 


.21864 


4.57363 


.23700 


4.21933 


.25552 


3.91364 


.27419 


3.64705 


40 


21 


.21895 


4.56726 


.23731 


4.21387 


.25583 


3.90890 


.27451 


3.64289 


39 


22 


.21925 


4.56091 


.23762 


4.20342 


.25614 


3.90417 


.27482 


3.63874 


33 


23 


.21956 


4.55458 


.23793 


4.20298 


.25645 


3.89945 


.27513 


3.63461 


37 


24 


.21986 


4.54826 


.23823 


4.19756 


.25676 


3.89474 


.27545 


3.63048 


36 


25 


.22017 


4.54196 


.23854 


4.19215 


.25707 


3.89004 


.27576 


3.62636 


&5 


26 


.22047 


4.53568 


.23885 


4.18675 


.25738 


3.88536 


.27607 


3.62224 


34 


27 


.22078 


4.52941 


.23916 


4.18137 


.25769 


3.88068 


.27638 


3.61814 


33 


28 


.22108 


4.52316 


.23946 


4.17600 


.25800 


3.87601 


.27670 


3.61405 


32 


29 


.22139 


4.51693 


.23977 


4.17064 


.25831 


3.87136 


.27701 


3.60996 


31 


30 


.22169 


4.51071 


.24008 


4.16530 


.25862 


3.86671 


.27732 


3 60588 


30 


31 


.22200 


4.50451 


.24039 


4.15997 


.25S93 


3.86208 


.27764 


3.60181 


29 


32 


.22231 


4.49832 


.24069 


4.15465 


.25924 


3.85745 


.27795 


3.59775 


23 


33 


.22261 


4.49215 


.24100 


4.14934 


.25955 


3.8528-1 


.27826 


3.59370 


27 


34 


.22292 


4.48600 


.24131 


4.14405 


.25986 


3.84824 


.27858 


3.58966 


26 


35 


.22322 


4.47986 


.24162 


4.13877 


.26017 


3.84364 


.27889 


3.58562 


25 


36 


.22353 


4.47374 


.24193 


4.13350 


.26048 


3.83906 


.27921 


3.58160 


24 


37 


.22383 


4.46764 


.24223 


4.12825 


.26079 


3.83449 


.27952 


3.57758 


23 


38 


.22414 


4.46155 


.24254 


4.12301 


.26110 


3.82992 


.27983 


3.57357 


22 


39 


.22444 


4.45548 


.24285 


4.11778 


.26141 


3.82537 


.28015 


3.56957 


21 


40 


.22475 


4.44942 


.24316 


4.11256 


.26172 


3.82083 


.28046 


3.56557 


20 


41 


.22505 


4.44338 


.24347 


4.10736 


.26203 


3.81630 


.28077 


3.56159 


19 


42 


.22536 


4.43735 


.24377 


4.10216 


.26235 


3.81177 


.28109 


3.55761 


18 


43 


.22567 


4.43134 


.24408 


4.09699 


.26266 


3.80726 


.28140 


3.55364 


17 


44 


.22597 


4.42534 


.24439 


4.09182 


.26297 


3.80276 


.28172 


3.54968 


16 


45 


.22628 


4.41936 


.24470 


4.08666 


.26328 


3.79827 


.28203 


3.54573 


15 


46 


.22858 


4.41340 


.24501 


4.08152 


.26359 


3.79378 


.28234 


3.54179 


14 


47 


.22689 


4.40745 


.24532 


4.07639 


.26390 


3.78931 


.28266 


3.53785 


13 


48 


.22719 


4.40152 


.24562 


4.07127 


.26421 


3.78485 


.28297 


3.53393 


12 


49 


.22750 


4.39560 


.24593 


4.06616 


.26452 


3.78040 


.28329 


3.53001 


11 


50 


.22781 


4.38969 


.24624 


4.06107 


.26483 


3.77595 


.28360 


3.52609 


10 


51 


.22811 


4.38381 


.24655 


4.05599 


.26515 


3.77152 


.28391 


3.52219 


9 


52 


.22842 


4.37793 


.24686 


4.05092 


.26546 


3.76709 


.28423 


3.51829 


8 


53 


.22872 


4.37207 


.24717 


4.04586 


.26577 


3.76268 


.28454 


3.51441 


7 


54 


.22903 


4.36623 


.24747 


4.04081 


.26608 


3.75828 


.28486 


3.51053 


6 


55 


.22034 


4.36040 


.24778 


4.03578 


.26639 


S. 75388 


.28517 


3.50666 


5 


56 


.22964 


4.35459 


.24809 


4.03076 


.26670 


3.74950 


.28549 


3.50279 


4 


57 


.22995 


4.34879 


.24840 


4.02574 


.26701 


3.74512 


.28580 


3.49894 


3 


58 


.23026 


4.34300 


.24871 


4.02074 


.26733 


3.74075 


.28612 


3.49509 


2 


59 


.23056 


4.33723 


! .24902 


4.01576 


.26764 


3.73640 


.28643 


3.49125 


1 


CO 

/ 


.23087 


4.33148 


| .24933 


4.01078 


.26795 


3.73205 


.28675 


3,48741 




/ 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 

7 


Tang 
4° 


77° 


76° 


; 7 


5° ! 



213 



NA TUBAL TANGENTS AND COTANGENTS. 



1 

~0 


16° 


17° 


18° 


19° 


/ 
60 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 
.34433 


Cotang 


.28675 


3.48741 I 


.30573 


3.27085 


.32492 


3.07768 


2.90421 


1 


.28706 


3.48359 


.30605 


3.26745 


.32524 


3.07464 


.34465 


2.90147 


59 


2 


.28738 


3.47977 


.30637 


3.26406 


! .32556 


3.07160 


.34498 


2.89873 


58 


3 


.28769 


3.47596 


.30669 


3.26067 


.32588 


3.06857 


.34530 


2.89600 


57 


4 


.28800 


3.47216 


.30700 


3.25729 


.32621 


3.06554 


.34563 


2.89327 


56 


5 


.28832 


3.46837 


.30732 


3.25392 


.32653 


3.06252 


.34596 


2.89055 


55 


6 


.28864 


3.46458 


.30764 


3.25055 


.32685 


3.05950 


.34628 


2.88783 


54 


7 


.28895 


3.46080 


.30796 


3.24719 


.32717 


3.05649 


.34661 


2.88511 


53 


8 


.28927 


3.45703 


.30828 


3.24383 


.32749 


3.05349 


.34693 


2.88240 


52 


9 


.28958 


3.45327 


.30860 


3.24049 


1 .32782 


3.05049 


.34726 


2.87970 


51 


10 


.28990 


3.44951 


.30891 


3.23714 


.32814 


3.04749 


.34758 


2 87700 


50 


11 


.29021 


3.44576 


.30923 


3.23381 


.32846 


3.04450 


.34791 


2.87430 


49 


12 


.29053 


3.44202 


.30955 


3.23048 


.32878 


3.04152 


.34824 


2.87161 


48 


13 


.29084 


3.43829 


.30987 


3.22715 


.32911 


3.03854 


.34856 


2.86892 


47 


14 


.29116 


3.43456 


.31019 


3.22384 


.32943 


3.03556 


.34889 


2.86624 


46 


15 


.29147 


3.43084 


.31051 


3.22053 


.32975 


3.03260 


.34922 


2.86356 


45 


16 


.29179 


3.42713 


.31083 


3.21722 


.33007 


3.02963 


.34954 


2.86089 


44 


17 


.29210 


3.42343 


.31115 


3.21392 


.33040 


3.02667 


.34987 


2.85822 


43 


13 


29242 


3.41973 


.31147 


3.21063 


.33072 


3.02372 


.35020 


2.85555 


42 


19 


.29274 


3.41604 


.31178 


3.20734 


.33104 


3.02077 


.35052 


2.85289 


41 


20 


.29305 


3.41236 


.81210 


3.20406 


.33136 


3.01783 


.35085 


2.85023 


40 


21 


.29337 


3.40869 


.31242 


3.20079 


.33169 


3.01489 


.35118 


2.84758 


39 


22 


.29368 


3.40502 


.31274 


3.19752 


.33201 


3.01196 


.35150 


2.84494 


38 


23 


.29400 


3.40136 


.31306 


3.19426 


.33233 


3.00903 


.35183 


2.84229 


37 


24 


.29432 


3.39771 


.31338 


3.19100 


.33266 


3.00611 


.35216 


2.83965 


36 


25 


.29463 


3.39406 


.31370 


3.18775 


.33298 


3.00319 


.35248 


2.83702 


35 


26 


.29495 


3.39042 


.31402 


3.18451 


.33330 


3.00028 


.35281 


2.83439 


34 


27 


.29526 


3.38679 


.31434 


3.18127 


.33363 


2.99738 


.35314 


2.83176 


33 


28 


.29558 


3.38317 


.31466 


3.17804 


.33395 


2.99447 


.35346 


2.82914 


32 


29 


.29590 


3.37955 


.31498 


3.17481 


.33427 


2.99158 


.35379 


2.82653 


31 


30 


.29621 


3.37594 


.31530 


3.17159 


.33460 


2.98868 


.35412 


2.82391 


30 


31 


.29653 


3.37234 


.31562 


3.16838 


.33492 


2.98580 


.35445 


2.82130 


29 


32 


.29685 


3.36875 


.31594 


3.16517 


.33524 


2.98292 


.35477 


2.81870 


28 


33 


.29716 


3.36516 


.31626 


3.16197 


.33557 


2.98004 


.35510 


2.81610 


27 


34 


.29748 


3.36158 


.31658 


3.15877 


.33589 


2.97717 


.35543 


2.81350 


26 


35 


.29780 


3.35800 


.31690 


3.15558 


.33621 


2.97430 


.35576 


2.81091 


25 


36 


.29811 


3.35443 


.31722 


3.15240 


.33654 


2.97144 


.35608 


2.80833 


24 


37 


.29843 


3.3508? 


.31754 


3.14922 


.33686 


2.96858 


.35641 


2.80574 


23 


38 


.29875 


3.34732 


.31786 


3.14005 


.33718 


2.96573 


.35674 


2.80316 


22 


39 


.29906 


3.31377 


.31818 


3.14288 


.33751 


2.96288 


.35707 


2.80059 


21 


40 


.29938 


3.34023 


.31850 


3.13972 


.33783 


2.96004 


.35740 


2.79802 


20 


41 


.29970 


3.33670 


.31882 


3.13656 


.33816 


2.95721 


.35772 


2.79545 


19 


42 


.30001 


3.33317 


.31914 


3.13341 


.33848 


2.95437 ! 


.35805 


2.79289 


18 


43 


.30033 


3.32965 


.31946 


3.13027 


.33881 


2.95155 


.35838 


2.79033 


17 


44 


.30065 


3.32614 


.31978 


3.12713 


.33913 


2.94872 


.35871 


2.78778 


16 


45 


.30097 


3.32264 


.32010 


3.12400 


.33945 


2.94591 


.35904 


2.78523 


15 


46 


.30123 


3.31914 


.32042 


3.12087 


.33978 


2.94309 


.35937 


2.78269 


14 


47 


.30160 


3.31565 


.32074 


3.11775 


.34010 


2.94028 


.35969 


2.78014 


13 


48 


.30192 


3.31216 


.32106 


3.11464 


.34043 


2.93748 


.36002 


2.77761 


12 


49 


.30224 


3.30868 


.32139 


3.11153 


.34075 


2.93468 


.36035 


2.77507 


11 


50 


.30255 


3.30521 


.32171 


3.10842 


.34108 


2.93189 


.36068 


2.77254 


10 


51 


.30287 


3.30174 


.32203 


3.10532 


.34140 


2.92910 


.36101 


2.77002 


9 


52 


.30319 


3.29829 


.32235 


3.10223 


.34173 


2.92632 


.36134 


2.76750 


8 


53 


.30351 


3.29483 


.32267 


3.09914 


.34205 


2.92354 


.36167 


2.76498 


7 


54 


.30382 


3.29139 


.32299 


3.09606 


.34238 


2.92076 


.36199 


2.76247 


6 


55 


.30414 


3.28795 


.32331 


3.09298 


.34270 


2.91799 


.36232 


2.75996 


5 


56 


.30446 


3.28452 


.32363 


3.08991 


.34303 


2.91523 


.36265 


2.75746 


4 


57 


.30178 


3.28109 


.32396 


3.08685 


.34335 


2.91246 


.36298 


2.75496 


3 


58 


.30509 


3.27767 


.32428 


3.08379 


.34368 


2.90971 


.36331 


2.75246 


2 


59 


.30541 


3.27426 


.32460 


3.08073 


.34400 


2.90696 


.36364 


2.74997 


1 


60 
/ 


.30573 
Cotang 


3.270S5 


.32492 


3.07768 


.34433 


2.90421 


.36397 


2.74748 



ft 


Tang 


Cotang 


Tang 


Cotang 


Tang 


Cotang 


Tang 


73° 


72° 


71° 


70* 



214 



